JEE Main · 2023 · 29 Jan 2023 · Shift 2 · Circle

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

Answer: 11

Solution

The circle is

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

. Since

P Q

is the chord of contact of the tangents from

S (α, β)

, its equation

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

must be the same line as

x + y - 3 = 0

. Comparing coefficients,

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

. The first equality gives

α = β - 1

, and pairing the first ratio with the third gives

3 (α - 2) = 2 α + 3 β + 3

, that is

α = 3 β + 9

. Solving the two,

β = - 5

and

α = - 6

. Therefore

4 α - 7 β = 4 (- 6) - 7 (- 5) = 11

JEE Main · 2023 · 29 Jan 2023 · Shift 2 · Integer answer

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