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Circle

Section: Mathematics

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Question : 17 of 89

Marks: +1, -0

If the pole of the line

x+2by-5=0

with respect to the circle

S \equiv x^2+y^2-4x-6y+4=0

lies on the line

x+by+1=0

, then the polar of the point (

b, -b

) with respect to the circle

S=0

[AP EAPCET 23 May 2025 S2]

$\;5y-6=0$
$\; y-6=0$
$\; x+5y-6=0$
$\; 5x+y-6=0$

Solution:

Let

(h, k)

be the pole of the line

x+2by-5=0

with respect to the circle

\; s \equiv x^2+y^2-4x-6y+4=0

\; \therefore \; \; hx+ky-2(x+h)-3(y+k)+4=0

\; (h-2)x+(k-3)y-2h-3k+4=0

is coincide with

x+2by-5=0

\; \frac{h-2}{1} \; = \; \frac{k-3}{2b} = \; \frac{2h+3k-4}{5}

\Rightarrow \; \frac{h-2}{1} \; = \; \frac{k-3}{2b}

2bh - k = 4b - 3 \; \; \; \cdot\cdot\cdot\cdot\cdot\cdot\cdot (i)

and

\; \frac{h-2}{1} = \; \frac{2h+3k-4}{5}

h-k=2 \; \; \; \cdot\cdot\cdot\cdot\cdot\cdot\cdot (ii)

Solving Eqs. (i) and (ii), we get

h = \; \frac{4b-5}{2b-1}, \; k = \; \frac{-3}{2b-1}

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