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

Question

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 (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

The given line is polar or P(2, β) w.r.t. given circle x2+y2-4 x-6 y-3=0 Chord or contact α x +β y -2( x +α)-3( y +β)-3=0 ⇒(α-2) x +(β-3) y -(2 α+3 β+3)=0....(i) ∵ But the equation of chord of contact is given as: x+y-3=0...(ii) comparing the coefficients (α-2/1)=(β-3/1)=-((2 α+3 β+3/-3)) On solving α=-6 β=-5 Now 4 α-7 β=11

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 4α−7β is equal to

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