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 , then 4 - 7 is equal to ______ . [2023]

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Solution

Q.
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 $S (α, β)$ , then $4 α - 7 β$ is equal to ______ . [2023]

Ans.
(11)

$The equation of circle can be written as,$

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

$Chord of contact at (α, β) is$

$\Rightarrow α x + β y - 2 (x + α) - 3 (y + β) - 3 = 0$

$\Rightarrow (α - 2) x + (β - 3) y - (2 α + 3 β + 3) = 0 ...(i)$

$∵$ $But the equation of chord of contact is given:$

$x + y - 3 = 0 ...(ii)$

$Comparing equations (i) and (ii):$

$\frac{α - 2}{1} = \frac{β - 3}{1} = - (\frac{2 α + 3 β + 3}{- 3})$

$On solving, α = - 6, β = - 5$

$So, 4 α - 7 β = 4 (- 6) - 7 (- 5) = 11$

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