Question

Find the pole of the line 3x + 5y + 17 = 0 with respect to the circle x² + y² + 4x + 6y + 9 = 0

Answer 1

Given circle is x² + y² + 4x + 6y + 9 = 0 ............(i) 

and given line is 3x + 5y + 17 = 0 ............(ii) 

Let P(α, β) be the pole of line (ii) with respect to circle (i) 

Now equation of polar of point P(α, β) with respect to circle (i) is 

xα + yβ + 2(x + α) + 3(y + β) + 9 = 0 

or (α + 2)x + (β + 3) y + 2α + 3β + 9 = 0 ............(iii) 

Now lines (ii) and (iii) are same, therefore,

From (i) and (ii), we get 

5α + 10 = 3β + 9 or 5α – 3β = – 1 ............(iv) 

From (i) and (iii), we get 

17α + 34 = 6α + 9β + 27 or 11α – 9β = –7 ............(v) 

Solving (iv) & (v), we get α = 1, β = 2 

Hence required pole is (1, 2).