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Question

The equation of the image of the circle x2+y2+16x24y+183=0 along the line mirror 4x+7y+13=0 is:

A
x2+y2+32x4y+235=0
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B
x2+y2+32x+4y235=0
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C
x2+y2+32x4y235=0
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D
x2+y2+32x+4y+235=0
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Solution

The correct option is D x2+y2+32x+4y+235=0
The given equation of the circle is x2+y2+16x24y+183=0, which can be written as,

(x+8)2+(y12)2=(5)2

Hence we can see the center of the circle , let's say O(8,12) and radius of the circle is r=5

If we mirror the image of the given circle by the line 4x+7y+13=0, the radius of the circle won't change. But the position of center will get change. Let's assume the new center will be O(α,β)

By using below equation to find O(α,β),

α(8)4=β127=2(4(8)+7(12)+13)42+72

α(8)4=β127=2

α=16,β=2

Hence new center O is O(16,2)

The equation of the image of the circle through the mirror will be,

(x+16)2+(y+2)2=(5)2

x2+y2+32x+4y+235

So correct option is D

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