Chapter 9 Analytic Geometry

Distance and Midpoint Formulas 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 c2 = a2 + b2

Example Find the distance between point D and point F.

Distance Formula D = √(x2 – x1)2 + (y2 – y1)2

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

The circle with center (h,k) and radius r has the equation Equation of a Circle The circle with center (h,k) and radius r has the equation (x – h)2 + (y – k)2 = r2

Example Find an equation of the circle with center (-2,5) and radius 3. (x + 2)2 + (y – 5)2 = 9

Translation Sliding a graph to a new position in the coordinate plane without changing its shape

Translation

Example Graph (x – 2)2 + (y + 6)2 = 4

Example If the graph of the equation is a circle, find its center and radius. x2 + y2 + 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 = a(x-h)2 x - h = a(y-k)2 Vertex (h,k) symmetry x = h x - h = a(y-k)2 Vertex (h,k) symmetry y = k

(h,k) is the vertex of the parabola Equation of a Parabola Remember: y – k = a(x – h)2 (h,k) is the vertex of the parabola

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)

Example 1

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 – h)2 a = 1/4c where c is the distance between the vertex and focus Parabola opens upward if a>0, and downward if a< 0

y – k = a(x – h)2 Vertex (h, k) Focus (h, k+c) Directrix y = k – c Axis of Symmetry x = h

x - h = a(y –k)2 a = 1/4c where c is the distance between the vertex and focus Parabola opens to the right if a>0, and to the left if a< 0

x – h = a(y – k)2 Vertex (h, k) Focus (h + c, k) Directrix x = h - c Axis of Symmetry y = k

Example 3 Find the vertex, focus, directrix , and axis of symmetry of the parabola: y2 – 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

Ellipse- major y-axis

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 denominator of y2 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 equation b2 =a2 – c2

Example 2 Graph the ellipse 4x2 + y2 = 64 and find its foci

Example 3 Find an equation of an ellipse having x-intercepts √2 and - √2 and y-intercepts 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 x-intercepts and no y-intercepts (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. x2 – 4y2 – 2x – 16y – 11 = 0