10.4 Hyperbolas

The standard form with center (h,k) is Note: a is under the positive term. It is not necessarily true that a is bigger than b.

Let’s take a look at the first hyperbola form. V(h-a,k) c V(h+a,k) b b C(h,k) F(h-c,k) F(h+c,k) a c is the distance from the center to the foci. Note: If c is the distance from the center to F, and all radii of a circle = , then the hyp. of the right triangle is also c. Therefore, to find c, a2 + b2 = c2

Sketch the hyperbola whose equation is 4x2 - y2 = 16. First divide by 16. Write down a, b, c and the center pt. a = 2 b = 4 Note: a is always under the (+) term. C(0,0) Now find c. Let’s sketch the hyperbola.

F F V V Now, we need to find the equations of the asymptotes. What are their slopes and one point that is on both lines?

Sketch the graph of 4x2 - 3y2 + 8x +16 = 0 4(x2 + 2x ) - 3y2 = -16 +1 + 4 4(x + 1)2 - 3y2 = -12 Now, divide by -12 and switch the x and y terms. C( , ) a = b = c = e = Sketch

V( , ) ( , ) F( , ) ( , ) F V V Eq. of asymptotes. F

Classifying a conic from its general equation. Ax2 + Cy2 + Dx + Ey + F = 0 If: A = C AC = 0 , A = 0 or C = 0, but not both AC > 0, AC < 0 Both A and C = 0