The equation of the given circle is:
x2+y2+8x−16y+64=0
We can complete the square to rewrite the equation:
x2+8x+16+(y2−16y+64)=16
This simplifies to:
(x+4)2+(y−8)2=42
Clearly, its center is (−4,8) and its radius is 4.
To find the equation of the image of the circle in the line mirror x=0, we reflect the center (−4,8) across the y-axis. The reflected center will be (4,8).
So, the equation of the image is:
(x−4)2+(y−8)2=42
Expanding this:
x2−8x+16+y2−16y+64=16
Simplifying:
x2+y2−8x−16y+64=0
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