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Question: Answered & Verified by Expert
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
MathematicsCircleJEE MainJEE Main 2023 (29 Jan Shift 2)
Solution:
1857 Upvotes Verified Answer
The correct answer is: 11

A Circle having the centre 2,3 and radius 4.

So, the equation of circle will be 

x-22+y-32=16

x2+y2-4x-6y-3=0

Now, it is given x+y=3 passes through two points P & Q on circle and tangents are drawn at P & Q intersect at Sα,β,

So, x+y=3 will be chord of contact.

Now equation of chord of contact will be 

Tα,β=0

αx+βy-2x+α-3y+β-3=0

xα-2+yβ-3-2α-3β-3=0

Now comparing above equation with x+y-3=0, we get

α-21=β-31=2α+3β+33

Now on solving above relation we get, 

α=-6 & β=-5

Hence, the value of 

4α-7β=-24+35=11

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