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Chapter 9 Analytic Geometry Section 9-1 Distance and Midpoint Formulas Pythagorean Theorem If the length of the hypotenuse of a right triangle is c, and the lengths of the other two sides are a and b, then 2 2 2 c =a +b Example 8 6 D 4 2 E -5 F 5 -2 Find the distance between point D and point F. 10 Distance Formula D = √(x2 – 2 x1) + (y2 – 2 y1) Example Find the distance between points A(4, -2) and B(7, 2) d = 5 Midpoint Formula M( x1 + x2, y1 + y2) 2 2 Example Find the midpoint of the segment joining the points (4, -6) and (-3, 2) M(1/2, -2) Section 9-2 Circles Conics Are obtained by slicing a double cone Circles, Ellipses, Parabolas, and Hyperbolas Equation of a Circle The circle with center (h,k) and radius r has the equation 2 2 2 (x – h) + (y – k) = r Example Find an equation of the circle with center (-2,5) and radius 3. 2 2 (x + 2) + (y – 5) = 9 Translation Sliding a graph to a new position in the coordinate plane without changing its shape Translation 8 6 4 2 -10 -5 5 -2 10 Example 6 4 Graph (x – -10 2 2) 2 -5 + (y + 5 -2 -4 -6 -8 -10 2 6) 10 =4 Example If the graph of the equation is a circle, find its center and radius. 2 2 x + y + 10x – 4y + 21 = 0 Section 9-3 Parabolas Parabola A set of all points equidistant from a fixed line called the directrix, and a fixed point not on the line, called the focus Vertex The midpoint between the focus and the directrix. Parabola - Equations y-k = 2 a(x-h) Vertex (h,k) symmetry x x-h= =h 2 a(y-k) Vertex (h,k) symmetry y =k Equation of a Parabola Remember: y – k = a(x – (h,k) is the vertex of the parabola 2 h) Example 1 The vertex of a parabola is (-5, 1) and the directrix is the line y = -2. Find the focus of the parabola. (-5 4) 8 Example 1 6 4 2 Vertex (- 5,1) -5 dir ectrix (y = -2) 5 -2 -4 Example 2 Find an equation of the parabola having the point F(0, -2) as the focus and the line x = 3 as the directrix. y – k = a(x – a) a 2 h) = 1/4c where c is the distance between the vertex and focus b) Parabola opens upward if a>0, and downward if a< 0 y – k = a(x – c) Vertex 2 h) (h, k) d) Focus (h, k+c) e) Directrix y = k – c f) Axis of Symmetry x = h x - h = a(y a) a 2 –k) = 1/4c where c is the distance between the vertex and focus b) Parabola opens to the right if a>0, and to the left if a< 0 x – h = a(y – c) Vertex 2 k) (h, k) d) Focus (h + c, k) e) Directrix x = h - c f) Axis of Symmetry y = k Example 3 Find the vertex, focus, directrix , and axis of symmetry of the parabola: 2 y – 12x -2y + 25 = 0 Example 4 Find an equation of the parabola that has vertex (4,2) and directrix y = 5 Section 9-4 Ellipses Ellipse The set of all points P in the plane such that the sum of the distances from P to two fixed points is a given constant. Focus (foci) Each fixed point Labeled as F1 and F2 PF1 and PF2 are the focal radii of P Ellipse- major x-axis drag Ellipse- major y-axis drag Example 1 Find the equation of an ellipse having foci (-4, 0) and (4, 0) and sum of focal radii 10. Use the distance formula. Example 1 - continued Set up the equation PF1 + PF2 = 10 √(x + 4)2 + y2 + √(x – 4)2 + y2 = 10 Simplify to get x2 + y2 = 1 25 9 Graphing The graph has 4 intercepts (5, 0), (-5, 0), (0, 3) and (0, -3) Symmetry The ellipse is symmetric about the x-axis if the denominator of x2 is larger and is symmetric about the y-axis if the 2 denominator of y is larger Center The midpoint of the line segment joining its foci General Form x2 + y2 = 1 a2 b2 The center is (0,0) and the foci are (-c, 0) and (c, 0) where b2 = a2 – c2 focal radii = 2a General Form x2 + y2 = 1 b2 a2 The center is (0,0) and the foci are (0, -c) and (0, c) where b2 = a2 – c2 focal radii = 2a Finding the Foci If you have the equation, you can find the foci by solving the 2 2 2 equation b =a – c Example 2 Graph the ellipse 2 2 4x + y = 64 and find its foci Example 3 Find an equation of an ellipse having x-intercepts √2 and - √2 and yintercepts 3 and -3. Example 4 Find an equation of an ellipse having foci (-3,0) and (3,0) and sum of focal radii equal to 12. Section 9-5 Hyperbolas Hyperbola The set of all points P in the plane such that the difference between the distances from P to two fixed points is a given constant. Focal (foci) Each fixed point Labeled as F1 and F2 PF1 and PF2 are the focal radii of P Example 1 Find the equation of the hyperbola having foci (-5, 0) and (5, 0) and difference of focal radii 6. Use the distance formula. Example 1 - continued Set up the equation PF1 - PF2 = ± 6 √(x + 5)2 + y2 - √(x – 5)2 + y2 = ± 6 Simplify to get x2 - y2 = 1 9 16 Graphing The graph has two xintercepts and no yintercepts (3, 0), (-3, 0) Asymptote(s) Line(s) or curve(s) that approach a given curve arbitrarily, closely Useful guides in drawing hyperbolas Center Midpoint of the line segment joining its foci General Form x2 - y2 = 1 a2 b2 The center is (0,0) and the foci are (-c, 0) and (c, 0), and difference of focal radii 2a where b2 = c2 – a2 Asymptote Equations y = b/a(x) and y = - b/a(x) General Form y2 - x2 = 1 a2 b2 The center is (0,0) and the foci are (0, -c) and (0, c), and difference of focal radii 2a where b2 = c2 – a2 Asymptote Equations y = a/b(x) and y = - a/b(x) Example 2 Find the equation of the hyperbola having foci (3, 0) and (-3, 0) and difference of focal radii 4. Use the distance formula. Example 3 Find an equation of the hyperbola with asymptotes y = 3/4x and y = -3/4x and foci (5,0) and (-5,0) Section 9-6 More on Central Conics Ellipses with Center (h,k) • Horizontal major axis: (x –h)2 + (y-k)2 = 1 a2 b2 Foci at (h-c,k) and (h + c,k) where c2 = a2 - b2 Ellipses with Center (h,k) • Vertical major axis: (x –h)2 + (y-k)2 = 1 b2 a2 Foci at (h, k-c) and (h,c +k) where c2 = a2 - b2 Hyperbolas with Center (h,k) • Horizontal major axis: (x –h)2 - (y-k)2 = 1 a2 b2 Foci at (h-c,k) and (h + c,k) where c2 = a2 + b2 Hyperbolas with Center (h,k) • Vertical major axis: (y –k)2 - (x-h)2 = 1 a2 b2 Foci at (h, k-c) and (h, k+c) where c2 = a2 + b2 Example 1 Find an equation of the ellipse having foci (-3,4) and (9, 4) and sum of focal radii 14. Example 2 Find an equation of the hyperbola having foci (-3,-2) and (-3, 8) and difference of focal radii 8. Example 3 Identify the conic and find its center and foci, graph. 2 2 x – 4y – 2x – 16y – 11 = 0