A line is drawn through a fixed point P $$($$α, β$$)$$ to cut the circle $$x2+y2=r2$$ at A and B .Then PA⋅PB is equal to

Question

A line is drawn through a fixed point P $$($$α, β$$)$$ to cut the circle  $$x2+y2=r2$$  at A and  B .Then PA⋅PB is equal to

Infinity Learn · Accepted Answer

The equation of any line throughP $$($$α, β$$)$$ is x−α$$cos$$⁡θ$$=y$$−β$$sin$$⁡θ$$=k(say)Any$$ point on this line is  $$($$α$$+$$ k $$cos$$ θ, β $$+$$ k $$sin$$θ$$)$$. This point lies on the given circle $$if($$α$$+kcos$$⁡θ$$)2+($$β$$+ksin$$⁡θ$$)2=r2$$⇒$$k2+2k($$α$$cos$$⁡θ$$+$$β$$sin$$⁡θ$$)+$$α$$2+$$β$$2$$−$$r2=0$$             (i)This$$ equation, being quadratic in k, gives two values of k and hence the distances of two points A and B on the circle from the point P. Let $$PA=k1$$,$$PB=k2$$, where $$k1$$,$$k2$$ are the roots of equation $$(i)Then$$,  PAPB $$=k1k2=$$α$$2+$$β$$2$$−$$r2ALITER$$  PAPB is the power of the point $$P($$α,β$$) $$)$$ with respect to the circle $$x2+y2=r2$$ . Therefore , $$PAPB=$$α$$2+$$β$$2$$−$$r2$$

A line is drawn through a fixed point $P (α, β)$ to cut the circle
$x^{2} + y^{2} = r^{2}$ at $A$ and $B$ .Then $P A \cdot P B$ is equal to

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A line is drawn through a fixed point P (α, β) to cut the circle x2+y2=r2 at A and B .Then PA⋅PB is equal to

Remember concepts with our Masterclasses.

Ready to Test Your Skills?

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A line is drawn through a fixed point $P (α, β)$ to cut the circle
$x^{2} + y^{2} = r^{2}$ at $A$ and $B$ .Then $P A \cdot P B$ is equal to