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Chapter 10: Q. 9 (page 698)

Identify each equation. If it is a parabola, give its vertex, focus, and directrix; if it is an ellipse, give its center, vertices, and foci; if it is a hyperbola, give its center, vertices, foci, and asymptotes.

4x2-16x+16y+32=0

Short Answer

Expert verified

The vertex is 2,-1, focus is 2,-2 and directrix is y=0.

Step by step solution

01

Step 1. Given Information 

We are given the equation 4x2-16x+16y+32=0.

We need to find the type of conic.

02

Step 2. Determining the conic 

Rewriting the equation we get:

4x2-16x+16y+32=0鈬�4(x2-4x)+16(y+2)=0鈬�(x2-4x)+4(y+2)=0鈬�(x2-4x)=-4(y+2)鈬�(x2-4x+4)=-4(y+2-1)鈬�(x-2)2=-4(y+1)

This is the equation of a parabola. Comparing it with the standard equation x-h2=-4a(y-k)we get:

h=2,-4a=-4鈬�a=1k=-1

03

Step 3. Finding vertex, focus, and directrix  

Vertex is h,k=2,-1.

Focus is h,k-a=2,-1-1=2,-2.

Directrix is

y=k+a鈬�y=-1+1鈬�y=0

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