wiz-icon
MyQuestionIcon
MyQuestionIcon
1
You visited us 1 times! Enjoying our articles? Unlock Full Access!
Question

Through a given point P(h,k) secants are drawn to cut the circle x2+y2=a2 at the points A1,B2;A2,B2;....;An,Bn.
Prove that
(PA1.PB1)=(PA2.PB2).....(PAn.PBn)=PQ2
where PQ is the length of tangent from P to the given circle.

Open in App
Solution

PQ2=S1=h2+k2a2....(1)
Any secant through P(h,k) is xhcosθ=yksinθ=r
where r is the distance of P from any point on the line.
(rcosθ+h,rsinθ+k) is any point on the line.
If it lies on the circle, then
(rcosθ+h)2+(rsinθ+k)2=a2
or r2+2r(hcosθ+ksinθ)+(h2+k2a2)=0
It gives two values of r say PA and PB.
PA.PB=r1r2=h2+k2a2=S2
=PQ2=constant, by (1)
as it is independent of θ. Hence all the products are equal and each equals to PQ2.

flag
Suggest Corrections
thumbs-up
0
Join BYJU'S Learning Program
similar_icon
Related Videos
thumbnail
lock
Line and Ellipse
MATHEMATICS
Watch in App
Join BYJU'S Learning Program
CrossIcon