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A circle with centre 2 3 and radius 4 intersects the li... | null (Mathematics) | DivineJEE

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Mathematics

Exam

Mathematics

A circle with centre (2, 3) and radius 4 intersects the line $x+y=3$ at the points P and Q. If the tangents at P and Q intersect at the point $S(\alpha,\beta)$ , then $4\alpha-7\beta$ is equal to ___________.

Correct Answer:

11

Explanation

The line

x + y = 3

..... (i)

is polar of

S(\alpha ,\beta )

w.r.t. circle

{(x - 2)^2} + {(y - 3)^2} = 16

\Rightarrow {x^2} + {y^2} - 4x - 6y - 3 = 0

Equation of polar is

\alpha x + \beta y - 2(x + \alpha ) - 3(4 + \beta ) - 3 = 0

(\alpha - 2)x + (\alpha - 3)y - (2\alpha + 3\beta + 3) = 0

..... (ii)

(i) and (ii) represent the same.

\therefore

{{\alpha - 2} \over 1} = {{\beta - 3} \over 1} = {{2\alpha + 3\beta + 3} \over 3}

\alpha - \beta + 1 = 0

\alpha - 3\beta - 9 = 0

\Rightarrow \alpha = - 6,\beta = - 5

4\alpha - 7\beta = 11

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