What is the main focus of Lecture Note 1 in MA1200?
Coordinate Geometry and Conic Sections.
How can the position of a point be described in polar coordinates?
By considering the distance from the origin (r) and the angle (θ) with the positive x-axis.
p.17
Ellipse: Definition and Equation
What is the center of the ellipse?
The midpoint between the two foci F1 and F2.
What is the compound angle formula for cosine?
cos(α + θ) = cos(α)cos(θ) - sin(α)sin(θ)
p.25
Parabola: Definition and Equation
What is the axis of symmetry in a parabola?
A line that divides the parabola into two mirror-image halves.
p.25
Parabola: Definition and Equation
What is the vertex of a parabola?
The point where the parabola changes direction.
p.11
Conic Sections Overview
What are conic sections?
Curves that result from intersecting a right circular cone with a plane.
p.22
Ellipse: Definition and Equation
What is the general equation of an ellipse centered at C = (h, k)?
The equation is (x - h)²/a² + (y - k)²/b² = 1.
p.22
Ellipse: Definition and Equation
In the general equation of an ellipse, what do 'a' and 'b' represent?
'a' represents the semi-major axis and 'b' represents the semi-minor axis.
p.17
Ellipse: Definition and Equation
Why is the constant in the ellipse equation chosen as 2a?
To make the equation of the ellipse look 'nicer'.
p.11
Classification of Conic Sections
What are the four types of conic sections?
Circle, Ellipse, Hyperbola, Parabola.
p.35
Conic Sections Overview
How is the equation of the hyperbola derived from the original equation?
By completing the square.
p.29
Parabola: Definition and Equation
What is the general vertex of a parabola?
The vertex can be at (h, k) instead of the origin (0, 0).
What are the two representations of coordinate systems?
Cartesian coordinates (x, y) and Polar coordinates (r, θ).
p.20
Ellipse: Definition and Equation
What points does the ellipse pass through?
The points (a, 0), (-a, 0), (0, b), (0, -b) which are called vertices.
p.18
Ellipse: Definition and Equation
What are the coordinates of the foci of the ellipse?
F1 = -c, 0 and F2 = c, 0.
p.41
Hyperbola: Definition and Equation
What does the equation y - ? = ±√(A/C)(x - ?) represent?
A pair of (unparallel) straight lines.
p.17
Ellipse: Definition and Equation
What are the fixed points in an ellipse called?
Foci (denoted as F1 and F2).
p.17
Ellipse: Definition and Equation
What is the defining property of points P in an ellipse?
P F1 + P F2 = constant = 2a.
p.41
Hyperbola: Definition and Equation
What happens to the equation when F' = 0?
It becomes A(x - ?)^2 + C(y - ?)^2 = 0, representing a pair of (unparallel) straight lines.
What happens when the x-axis and y-axis are rotated through a positive angle?
A new x' and y' axis is formed, creating a new coordinate system.
How is the coordinate P represented in the x, y coordinate system?
x = r cos(θ) + α, y = r sin(θ) + α.
p.12
Circle: Definition and Equation
What is the fixed distance from the center of a circle to any point on the circle called?
The fixed distance is called the radius and is denoted by r.
What are the coordinates of point P in the x', y' coordinate system?
x' = r cos(α), y' = r sin(α).
In which direction is the angle considered positive during the rotation of axes?
Anti-clockwise direction.
p.61
Classification of Conic Sections
What is the relationship between tan(2θ) and the coefficients A, B, and C?
tan(2θ) = (B / (A - C)) if A ≠ C.
p.34
Classification of Conic Sections
How can you determine if a conic section is a hyperbola from its equation?
By rearranging the equation into the standard form of a hyperbola.
p.23
Ellipse: Definition and Equation
What is the equation of the ellipse given in the example?
4x² + 9y² - 8x + 36y - 9 = 0.
p.39
Classification of Conic Sections
What is the general equation of a conic section?
𝐴𝑥² + 𝐶𝑦² + 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0.
What are the equations used to describe the motion of a moving particle in physics?
The pair of equations: x = f(t) and y = g(t), where t is a parameter.
What is the transformation formula for y in the context of the rotation?
y = x' sin 30° + y' cos 30° = (1/2)x' + (√3/2)y'.
p.55
Classification of Conic Sections
What is the general equation of a conic section?
A x² + Bxy + C y² + Dx + Ey + F = 0
p.25
Parabola: Definition and Equation
What is a parabola?
The set of all points in a plane that are equidistant from a fixed line (directrix) and a fixed point (focus).
p.12
Circle: Definition and Equation
What is the center of a circle called?
The center is called point C.
p.34
Hyperbola: Definition and Equation
What do the variables h and k represent in the hyperbola equation?
They represent the coordinates of the center C = (h, k).
What transformation is applied to the x-y axes in the given text?
The axes are rotated by 30° in an anti-clockwise direction.
p.29
Parabola: Definition and Equation
What is the equation of a parabola with a vertical axis of symmetry?
If the axis of symmetry is x = h, the equation is (x - h)² = 4a(y - k).
p.52
Conic Sections Overview
What is the initial equation after substituting the formulae?
5𝑥′² + 𝑦′² + 6𝑥′ + 𝑦′ − 𝑥′ + 𝑦′ + 5² − 𝑥′ + 𝑦′² − 18² + 2𝑥′ + 𝑦′ − 14 − 𝑥′ + 𝑦′ + 26 = 0.
p.52
Conic Sections Overview
What is the final form of the equation derived from the simplification?
𝑥′² + 4𝑦′² − 2𝑥′ − 16𝑦′ + 13 = 0.
What is the first step in converting Cartesian coordinates to polar coordinates?
Locate the points on the x-y plane.
p.32
Hyperbola: Definition and Equation
What are the vertices of the hyperbola?
A1 = (-a, 0) and A2 = (a, 0).
What coordinate system will be adopted in the remaining section of the lecture?
The Cartesian coordinates system.
p.12
Circle: Definition and Equation
What defines a circle in coordinate geometry?
A circle is a set of all points such that the distance between the point and a fixed point (the center) is always fixed.
p.44
Conic Sections Overview
Why is it important to rewrite the conic section equation into the form 𝑥−ℎ²/𝑝−𝑦−𝑘²/𝑞=1?
To address the existence of the term 𝐵𝑥𝑦.
p.37
Classification of Conic Sections
What technique can be used to classify conic sections from their equations?
Completing the square technique.
p.53
Ellipse: Definition and Equation
What is the significance of the coordinates (1, 2) in the context of the ellipse?
They represent the center of the ellipse.
p.24
Ellipse: Definition and Equation
What are the coordinates of the vertex (3, -5/2) related to?
It is one of the vertexes of the ellipse.
p.23
Ellipse: Definition and Equation
What is the standard form of the ellipse after rewriting?
(x - 1/2)²/7² + (y + 2)²/3² = 1.
What does θ represent in polar coordinates?
The directed angle between OP and the positive x-axis.
What transformation is applied to the coordinate system?
The x-axis and y-axis are rotated by 45° in a clockwise direction.
p.37
Classification of Conic Sections
What is the significance of rewriting conic section equations into standard form?
It allows for easy classification into circle, ellipse, parabola, or hyperbola.
p.20
Ellipse: Definition and Equation
Where are the foci located if the ellipse is oriented vertically?
At F1 = (0, c) and F2 = (0, -c) on the y-axis.
p.22
Ellipse: Definition and Equation
What does the center of the ellipse (h, k) indicate?
It indicates the coordinates of the center of the ellipse.
p.45
Classification of Conic Sections
How can one identify the conic section from the equation?
By sketching the graph of the conic section.
What coordinates are highlighted in red in the graph?
Coordinates in the x'y' plane.
How do you express the equation of line L1 in slope-intercept form?
From 3x - 2y + 5 = 0, we get y = (3/2)x + 5/2.
p.18
Ellipse: Definition and Equation
What is the derived equation of the ellipse after simplification?
a(x - c)^2 + y^2 = a^2 - xc.
p.24
Ellipse: Definition and Equation
What equation is represented by the coordinates given in the lecture note?
The equation of the ellipse.
p.2
Conic Sections Overview
What is the general equation of a conic section used for?
To classify and identify the conic section in the 2-D plane.
p.42
Classification of Conic Sections
What condition must be met for A and C in the equation of conic sections?
If A > 0, then A and C must have the same sign (both positive or both negative).
p.33
Hyperbola: Definition and Equation
What is the equation of the hyperbola when the foci are at F1 = (0, c) and F2 = (0, -c)?
The equation is given by y²/a² - x²/b² = 1, where b² = c² - a².
What is the 'x-coordinate' of a point P in Cartesian coordinates?
The directed distance from point P to the y-axis.
p.34
Hyperbola: Definition and Equation
What is the general form of the equation of a hyperbola?
The general form is (x - h)²/a² - (y - k)²/b² = 1 or (y - k)²/a² - (x - h)²/b² = 1.
p.44
Classification of Conic Sections
What does the term 𝐵𝑥𝑦 indicate about a conic section?
It indicates that the conic section is being rotated from its 'standard position'.
p.32
Hyperbola: Definition and Equation
What is the standard equation of a hyperbola with foci at (c, 0) and (-c, 0)?
The equation is x²/a² - y²/b² = 1.
p.34
Hyperbola: Definition and Equation
What is the first step to find the center and vertices of the hyperbola given the equation -x² + 4y² + 4x - 24y + 28 = 0?
Rearranging the equation into standard form.
p.39
Classification of Conic Sections
What condition must be met for classifying conic sections?
At least one of 𝐴 or 𝐶 must be non-zero.
What is the slope of line L1?
The slope m1 of L1 is 3/2.
p.26
Parabola: Definition and Equation
What does the variable 'a' represent in the context of the parabola?
The distance from the focus to the directrix.
What do the variables r, α, and θ represent in the context of the formulas?
r is the radius, α is the angle, and θ is the rotation angle.
p.19
Ellipse: Definition and Equation
What does 2a represent in the context of an ellipse?
The sum of distances from any point on the ellipse to the two foci.
p.42
Conic Sections Overview
What is the general form of the equation after completing the square for conic sections?
A(x - ?)² + C(y - ?)² = F' where (F' ≠ 0).
In the notation P = (a, b), what do 'a' and 'b' represent?
'a' is the x-coordinate and 'b' is the y-coordinate of point P.
p.42
Ellipse: Definition and Equation
What is the standard form of the equation for an ellipse?
x - h²/a² + y - k²/b² = 1.
What is a key technique for interchanging between Cartesian and Polar coordinates?
Using the relationships between the two systems.
p.15
Circle: Definition and Equation
What is the equation of the circle given in the example?
x² + y² + 8x - 10y - 8 = 0.
p.20
Ellipse: Definition and Equation
What is the equation of an ellipse when the foci lie on the y-axis?
x²/b² + y²/a² = 1, where b² = a² - c².
p.20
Circle: Definition and Equation
What happens to the equation of the ellipse when a = b?
It becomes x² + y² = a², which is the equation of a circle.
How do you convert polar coordinates to Cartesian coordinates?
Use the formulas x = r cos(θ) and y = r sin(θ).
p.37
Classification of Conic Sections
What is the first step in classifying a conic section given its equation?
Rewrite the equation into the standard form.
p.32
Hyperbola: Definition and Equation
What is the relationship between a, b, and c in the context of a hyperbola?
b² = c² - a², where a, b > 0.
What is the general form of the conic section equation after rotation?
A'x'^2 + C'y'^2 + D'x' + E'y' + F' = 0.
What transformation is applied to the coordinate system?
The x-axis and y-axis are rotated by 45° in an anti-clockwise direction.
p.59
Classification of Conic Sections
What is the first step in the classification procedure of conic sections?
Transform the equation using the transformation formulas.
p.63
Parabola: Definition and Equation
What happens to the parabola when a = 1?
The equation becomes y² - 4y + x + 3 = 0.
p.24
Ellipse: Definition and Equation
What is the significance of the coordinates (9/2, -2) in the context of the ellipse?
It is another vertex of the ellipse.
p.23
Ellipse: Definition and Equation
What are the semi-major and semi-minor axes of the ellipse?
Semi-major axis = 7, Semi-minor axis = 3.
What is the transformation formula for y in the new coordinates?
y = x' sin(-45°) + y' cos(-45°) = (1/2) - x' + y'.
What do the values f(b) and g(b) represent in parametric equations?
They represent the coordinates (x, y) at the endpoint of the motion.
What is the equation for 𝐶 ′ in terms of 𝐴, 𝐵, 𝐶, and 𝜃?
𝐶 ′ = 𝐴 sin²(𝜃) - 𝐵 sin(𝜃) cos(𝜃) + 𝐶 cos²(𝜃).
What transformation is needed to rewrite the conic section equation?
Transform into the form A'x'² + C'y'² + D'x' + E'y' + F' = 0 using rotation of axes.
p.54
Classification of Conic Sections
What does the equation become after choosing the appropriate 𝜃?
𝐴'𝑥'² + 𝐶'𝑦'² + 𝐷'𝑥' + 𝐸'𝑦' + 𝐹' = 0.
p.41
Hyperbola: Definition and Equation
What is the general form of the equation after completing the square for a hyperbola?
A(x - ?)^2 - C(y - ?)^2 = F' where (F' ≠ 0).
p.44
Conic Sections Overview
What is the general form of a conic section in 2-D?
𝐴𝑥² + 𝐵𝑥𝑦 + 𝐶𝑦² + 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0, where 𝐴, 𝐵, 𝐶, 𝐷, 𝐸, and 𝐹 are constants.
p.14
Circle: Definition and Equation
What is the equation of the circle with center (1, -1) and radius 2?
The equation is (x - 1)² + (y + 1)² = 2².
p.2
Classification of Conic Sections
What are the four types of conic sections?
Circle, Ellipse, Parabola, and Hyperbola.
p.37
Classification of Conic Sections
What type of conic section is represented by the equation 4x² - 16x + 25y² - 84 = 0?
This equation can be classified after completing the square.
p.18
Ellipse: Definition and Equation
What happens when you take the square of both sides in the ellipse equation derivation?
You expand the terms to derive the equation of the ellipse.
What are the two types of coordinate systems used in 2-D?
Cartesian coordinates and another unspecified type.
In which direction is θ measured if it is greater than 0?
Anti-clockwise direction.
p.52
Conic Sections Overview
What is the final equation after completing the square?
𝑥′ − 1/2)²/2² − (𝑦′ − 2)²/1² = 1.
p.54
Classification of Conic Sections
What is the general form of the equation for a conic section?
𝐴𝑥² + 𝐵𝑥𝑦 + 𝐶𝑦² + 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0.
p.67
Parabola: Definition and Equation
What is the significance of the coefficients in the parabola equations?
They determine the orientation and position of the parabola.
p.27
Parabola: Definition and Equation
What happens to the equations of parabolas when x and y are interchanged?
The parabola's orientation changes.
p.42
Classification of Conic Sections
What occurs in the degenerate case when F' = 0?
The conic section becomes a single point.
p.58
Classification of Conic Sections
What is the general form of a conic section equation?
Ax² + Cy² + Dx + Ey + F = 0.
p.13
Circle: Definition and Equation
What does the variable 'r' represent in the equation of a circle?
The radius of the circle.
p.29
Parabola: Definition and Equation
What is the equation of a parabola with a horizontal axis of symmetry?
If the axis of symmetry is y = k, the equation is (y - k)² = 4a(x - h).
p.16
Circle: Definition and Equation
How is the radius of the circle calculated?
r = CP = √((2 - (-2))^2 + (3 - 1)^2) = √20.
p.26
Parabola: Definition and Equation
What condition must be satisfied for a point P(x, y) to lie on the parabola?
PQ = PF, where Q is the foot of the perpendicular from P to the directrix.
p.16
Circle: Definition and Equation
What is the equation of the circle derived in the example?
x^2 + y^2 - 4x - 6y - 7 = 0.
p.36
Hyperbola: Definition and Equation
What is the general equation of a hyperbola?
𝑥 − ℎ²/a² − 𝑦 − 𝑘²/b² = 1 or 𝑦 − 𝑘²/a² − 𝑥 − ℎ²/b² = 1.
p.31
Hyperbola: Definition and Equation
What does b² represent in the context of hyperbolas?
b² = c² - a², where c > a.
What is the significance of the values f(a) and g(a) in parametric equations?
They represent the coordinates (x, y) at the starting point of the motion.
What is the equation of line L derived from its slope?
The equation is 2x + 3y - 10 = 0.
p.32
Hyperbola: Definition and Equation
What are the asymptotes of the hyperbola?
The lines y = (b/a)x and y = -(b/a)x.
p.31
Hyperbola: Definition and Equation
What is the significance of the values a and b in the hyperbola equation?
a represents the distance from the center to the vertices, and b represents the distance related to the asymptotes.
p.33
Hyperbola: Definition and Equation
What are the coordinates of the foci of the hyperbola?
The foci are F1 = (0, c) and F2 = (0, -c).
What trigonometric identity is used in the equation for 𝐵 ′?
2 sin(𝜃) cos(𝜃) = sin(2𝜃) and cos²(𝜃) − sin²(𝜃) = cos(2𝜃).
What is the compound angle formula for sine?
sin(α + θ) = sin(α)cos(θ) + cos(α)sin(θ)
p.15
Circle: Definition and Equation
What is the final form of the circle's equation after completing the square?
(x + 4)² + (y - 5)² = 49.
p.32
Hyperbola: Definition and Equation
What is the center of the hyperbola?
The origin (mid-point of the foci F1 and F2).
p.18
Ellipse: Definition and Equation
What is the significance of '2a' in the ellipse equation?
It represents the constant sum of distances from any point on the ellipse to the foci.
p.14
Circle: Definition and Equation
What is the expanded form of the equation of the circle with center (1, -1) and radius 2?
The expanded form is x² + y² - 2x + 2y = 0.
p.63
Parabola: Definition and Equation
What is the effect of setting a = 0 in the parabola's equation?
The equation changes, affecting the shape and position of the parabola.
p.33
Hyperbola: Definition and Equation
What is the center of the hyperbola described?
The center is at the origin (0, 0).
p.42
Ellipse: Definition and Equation
What happens when A ≠ C in the conic section equation?
The equation represents an ellipse.
What is the equation used to solve for 𝐵 ′?
𝐵 ′ = −2𝐴 cos(𝜃) sin(𝜃) + 𝐵 cos²(𝜃) − sin²(𝜃) + 2𝐶 sin(𝜃) cos(𝜃) = 0.
What is the classification technique for conic sections using rotation of axes?
Using the equation Ax² + Bxy + Cy² + Dx + Ey + F = 0.
p.22
Ellipse: Definition and Equation
What is the significance of the equation (x - h)²/a² + (y - k)²/b² = 1?
It describes the shape and position of an ellipse in a coordinate system.
p.14
Circle: Definition and Equation
How can the equation of a circle be expressed to find its center and radius?
In the form (x - h)² + (y - k)² = r².
p.21
Ellipse: Definition and Equation
What are the coordinates of the vertices of the ellipse?
(3, 0), (-3, 0), (0, 2), (0, -2).
p.32
Hyperbola: Definition and Equation
What does 2a represent in the context of a hyperbola?
It is the difference between the distances from any point on the hyperbola to the two foci.
p.51
Conic Sections Overview
What is the given equation of the conic section?
5x² + 6xy + 5y² - 18x - 14y + 26 = 0.
p.36
Classification of Conic Sections
How can the equations of conic sections be expressed in a general form?
𝐴𝑥² + 𝐶𝑦² + 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0.
How is the position of a point described in Cartesian coordinates?
By the directed distances from the point to the x-axis and y-axis.
p.63
Parabola: Definition and Equation
What is the vertex of the parabola when a = 1?
The vertex is at (h, k) = (1, 2).
p.31
Hyperbola: Definition and Equation
What condition must be satisfied for b² in hyperbolas?
b² must be greater than 0 (b² > 0).
What is the first step to find the equation of the conic section in x'y' coordinates?
Apply the transformation formulas for x and y to the original conic equation.
p.54
Classification of Conic Sections
What condition should be chosen for 𝜃 to simplify the conic section equation?
Choose 𝜃 such that 𝐵'𝑥'𝑦' = 0 or 𝐵' = 0.
What is the final simplified equation derived from the conditions?
𝐴 − 𝐶 sin²(𝜃) = 𝐵 cos²(𝜃).
p.36
Ellipse: Definition and Equation
What is the general equation of an ellipse?
𝑥 − ℎ²/a² + 𝑦 − 𝑘²/b² = 1.
What are the relations between polar coordinates (r, θ) and Cartesian coordinates (x, y)?
x = r cos(θ), y = r sin(θ), r = x² + y², tan(θ) = y/x.
p.52
Conic Sections Overview
What does the equation simplify to after combining like terms?
2𝑥′² + 8𝑦′² − 4𝑥′ − 32𝑦′ + 26 = 0.
p.29
Parabola: Definition and Equation
How can the general equation of a parabola be derived from the standard form?
By replacing x with (x - h) and y with (y - k) in the standard form.
What does r represent in polar coordinates?
The distance between the point P and the origin.
What visual technique is used to identify the conic section?
Rotating the viewpoint by a certain degree.
p.31
Hyperbola: Definition and Equation
What are the coordinates of the foci for the hyperbola in standard form?
F1 = (-c, 0) and F2 = (c, 0).
p.67
Parabola: Definition and Equation
What does the vertex of the parabola represent in the given equations?
The vertex is at the point (ℎ, 𝑘) = (0, -2).
p.33
Hyperbola: Definition and Equation
What are the coordinates of the vertices of the hyperbola?
The vertices are A1 = (0, -a) and A2 = (0, a).
p.59
Classification of Conic Sections
How is A' calculated in the transformation process?
A' = A cos² θ + B cos θ sin θ + C sin² θ.
p.67
Parabola: Definition and Equation
What is the vertex form of a parabola derived from the equations?
The vertex form is represented as 𝑦 = 𝑎(𝑥 - ℎ)² + 𝑘.
What is the transformation formula for x in the context of the rotation?
x = x' cos 30° - y' sin 30° = (√3/2)x' - (1/2)y'.
p.13
Circle: Definition and Equation
What is the general equation of a circle with center at C(h, k)?
(x - h)² + (y - k)² = r².
p.36
Parabola: Definition and Equation
What is the general equation of a parabola?
𝑥 − ℎ² = 4𝑎(𝑦 − 𝑘) or 𝑦 − 𝑘² = 4𝑎(𝑥 − ℎ).
p.59
Classification of Conic Sections
What is the general form of the conic section equation discussed?
Ax² + Bxy + Cy² + Dx + Ey + F = 0 (for B ≠ 0).
p.13
Circle: Definition and Equation
In the equation (x - h)² + (y - k)² = r², what do 'h' and 'k' represent?
The x and y coordinates of the center of the circle.
p.65
Conic Sections Overview
What is the equation of the given conic section?
3x² + 2√3xy + y² - x + 3y + 4 = 0.
What is the slope of line L if the slope of line L1 is 3/2?
The slope m of line L is -2/3.
p.67
Parabola: Definition and Equation
What shape does the parabola take in the 𝑥 ′ 𝑦 ′ - plane?
It is an inverted U-shape curve.
p.42
Circle: Definition and Equation
What type of conic section is represented when A = C?
The equation represents a circle.
p.63
Parabola: Definition and Equation
How is the vertex form of the parabola derived in the example?
By completing the square on the equation.
p.59
Classification of Conic Sections
What is the formula for B' in the transformed equation?
B' = -2A cos θ sin θ + B cos² θ - sin² θ + 2C sin θ cos θ.
What does the equation become when 𝐵 ′ = 0?
𝐴 ′ x ′² + 𝐶 ′ y ′² + 𝐷 ′ x ′ + 𝐸 ′ y ′ + 𝐹 ′ = 0.
What are the two basic coordinate systems?
Cartesian coordinates (xy-coordinate) and Polar coordinates (rθ-coordinate).
p.58
Conic Sections Overview
What should you know about the standard equations of conic sections?
You should know the standard equations for Circle, Ellipse, Parabola, and Hyperbola.
p.14
Circle: Definition and Equation
What is the center of the circle given by the equation x + 3² + y - 1² = 7?
The center is C = (-3, 1).
What are the transformation formulas for x and y in the new coordinate system?
x = (1/√2)x' - (1/√2)y', y = (1/√2)x' + (1/√2)y'.
p.59
Classification of Conic Sections
What transformation formulas are used to transform the conic section equation?
x = x' cos θ - y' sin θ and y = x' sin θ + y' cos θ.
What is the 'y-coordinate' of a point P in Cartesian coordinates?
The directed distance from point P to the x-axis.
What condition must be met to simplify the equation to 𝐵 ′ = 0?
Choose 𝜃 such that 𝐵 ′ = 0.
p.58
Classification of Conic Sections
How can you identify a conic section?
Using the completing square technique.
What is the transformation formula for x in the new coordinates?
x = x' cos(-45°) - y' sin(-45°) = (1/2)x' + y'.
p.59
Classification of Conic Sections
What is the alternative form of the conic section equation after transformation?
A'x'² + B'x'y' + C'y'² + D'x' + E'y' + F' = 0.
What transformation formulas are used to rewrite the conic section equation?
𝑥 = 𝑥' cos 𝜃 − 𝑦' sin 𝜃 and 𝑦 = 𝑥' sin 𝜃 + 𝑦' cos 𝜃.
p.33
Hyperbola: Definition and Equation
What are the coordinates of points B1 and B2 in relation to the hyperbola?
B1 = (-b, 0) and B2 = (b, 0).
p.58
Conic Sections Overview
What are the four conic sections?
Circle, Ellipse, Parabola, Hyperbola.
p.21
Ellipse: Definition and Equation
What are the coordinates of the foci of the ellipse?
F₁ = (-5, 0) and F₂ = (5, 0).
p.32
Hyperbola: Definition and Equation
What happens to the graph of the hyperbola as x and y get larger?
The graph approaches the asymptotes y = (b/a)x and y = -(b/a)x.
What is the condition for line L to be perpendicular to line L1?
The product of their slopes must equal -1.
p.54
Classification of Conic Sections
What is the first step to identify a conic section from its equation?
Use the transformation formula to rewrite the equation.
p.55
Classification of Conic Sections
What is the significance of the expression B² - 4AC?
It helps classify the type of conic section.
p.27
Parabola: Definition and Equation
What does 'a' represent in the equations of parabolas?
The distance from the vertex to the focus or directrix.
p.54
Classification of Conic Sections
How can one identify the conic section after rewriting the equation?
By using the results outlined in the reference material (P.39).
