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Question

Find the pole of the straight line 2x + y + 12 = 0 with respect to the circle x2+y24x+3y1=0.

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Solution

Polar of the point (x1,y1) with respect to the circle x2+y24x+3y1=0 is given by xx1+yy12x2x1+1.5y+1.5y11=0
Comparing the above equation with the equation of the straight line 2x+y+12=0, we have
x122=y1+1.51=2x1+1.5y1112
6x112+2x11.5y1+1=0
or 8x11.5y111=0 ...(1)
Also, 12y1+18+2x11.5y1+1=0 or 2x1+10.5y1+19=0 ...(2)
Multiplying equation (1) by 7 and adding that to equation (2), we have
58x1=58 or x1=1
y1=2
The pole is thus (1,2)

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