Conics Unit Ch. 8
Circles Equations of Circles The equation of a circle with center ( hk, ) and radius r units is ( x h) ( y k) r. Example 1: Write an equation of circle with center (8, 3) and radius 6. Example : Write an equation for a circle if the enpoints of a diameter are at ( 4,5) and (6, 3). Step 1: Use the Midpoint formula to find the center. x x y y ( hk, ), 1 1 Step : Use the center and an endpoint to find the raidus. The radius is. So r The equation of the circle is : Example 3: Write an equation of a circle with center ( 5,) and passes through ( 9,6).at Example 4: Write an equation of a circle with endpoints of a diameter at ( 4, ) and (8,4).
Tangent A line is tangent to a circle when it touches the circle at only one point. Example 4: Write an equation of a circle with center ( 6,5) tangent to the y-axis. Example 5: Write an equation of a circle with center (,8) tangent to y 4. ----------------------------------------------------------------------------------------------------------------------------- Example 6: Write an equation of the line tangent to the circle (1,4). x y 106 at the point
Example 7: Graphing Circles in Standard Form a) x y ( 1) ( 3) 9 b) x ( y 5) 4 Example 8: Use a completing the square technique to change circles equations into Standard Form. Then Graph the Circle. a) x y x y 10 8 16 0 b) x y x y 4 6 1
Ellipses Equations of Ellipses An ellipse is the set of all points in a plane such that the sum of the distances from two given points in the plane, called the foci, is constant. An ellipse has two axes of symmetry which contain the major and minor axes. In the table, the lengths a, b, and c are related by the formula: c a b Standard Form of Equation c is the distance between the focus and the center. ( x h) ( y k) ( y k) ( x h) 1 a b a b hk ( hk, ) Center (, ) Orientation Horizontal Vertical Foci ( h c, k) ( h c, k) ( h, k c) h k Vertices ( h a, k) ( h a, k) ( h, k a) h k Length of Major Axis a units a units Co-Vertices ( h, k b) ( h, k b) ( h b, k) h Length of Minor Axis b units b units Example 1: Write an equation for the ellipse graphed below. 1 (, c) (, a) ( b, k)
Example : Write an equation for the ellipse with vertices at ( 7,) and (5,), co-vertices at ( 1,0) and ( 1,4). Example 3: Write an equation of the ellipse with vertices at ( 8,4) and (4,4), foci at ( 3,4) and ( 1,4). Example 4: Write an equation of the ellipse with minor axis 6 units long and parallel to the x- axis, the major axis is 1 units long, and the center is at (6,1).
Example 5: Graphing Ellipses in Standard Form. Find, plot, and label coordinates of the center and foci. a) x y 1 b) y 1 ( x ) 9 16 4 5 1 Example 6: Use a completing the square technique to change ellipse equations into Standard Form. Then Graph the Ellipse. Find, label, and plot coordinates of the foci. b) x 4y 4y 3 b) 9x 6y 36x 1y 1
Hyperbolas Equations of Hyperbolas A hyperbola is the set of all points in a plane such that the absolute value of the difference of the distances from any point on the hyperbola to any two given points in the plane, called the foci, is constant. In the table, the lengths a, b, and c are related by the formula: c a b Standard Form of Equation c is the distance between the focus and the center. ( x h) ( y k) ( y k) ( x h) 1 a b a b hk ( hk, ) Center (, ) Orientation Horizontal Vertical Foci ( h c, k) ( h c, k) ( h, k c) h k Vertices ( h a, k) ( h a, k) ( h, k a) h k Length of Transverse Axis a units a units Length of Conjugate Axis b units b units Slope of Asymptotes *Asymptotes intersect at the center ( hk, ). b a 1 (, c) (, a) Example 1: Write an equation of the hyperbola with vertices at ( 7,0) and (7,0) and a conjugate axis of length 10. a b
Example : Write an equation for the hyperbola graphed below. Example : Write an equation for the hyperbola with vertices at (0,4) and (0, 4) and the equation of the asymptotes is 4 y x. 3
Example 5: Graphing Hyperbolas in Standard Form. Find, plot, and label the vertices, foci, and asymptotes. a) x y 1 b) 4 16 y ( x1) 16 9 1 Example 6: Use a completing the square technique to change a hyperbola equation into Standard Form. Then Graph the Hyperbola. a) 6y 4x 36y 8x 6
Parabolas Equations of Parabolas The standard form of the equation of a parabola with vertex (0,0) is as follows: Orientation Equation Focus Directrix Axis of Symmetry Vertical x 4py (0, p ) y p x 0 Horizontal y 4px ( p,0) x p y 0 1 8 Example 1: Graph x y Example : Write an equation of a parabola in standard form with focus at (0, 3).
Other Directrix Equations Classifying Conics Given a conic equation in General Form: Ax Bxy Cy Dx Ey F 0 B 4AC 0 and A C A C Ellipse Circle B 4AC 0 Hyperbola B 4AC 0 Parabola Example 1: Given in general form, classify the conic. a) 5x y 0x 4y 4 0 b) y 8x 1y 0 c) x y 4 18 18 0 d) x y x y 4 4 0
Solving Nonlinear Systems of Equations Example 1: Find the intersection points on the graph of the following two equations. x y 5 y x (Substitution Method) Example : Find the intersection points on the graph of the following two equations. x y x 3 8 x y 10 (Elimination Method) Example 3: Application The range of a radio station is bounded by a circle equation given by the following equation: x y 160 0 where the unit is miles. A straight highway can be modeled by the following equation: 1 y x 30. 3 Find the length of the highway that lies within the range of the radio station.