If p₁ and p₂ are the two values of p for which two perpendicular tangents can be drawn from the origin to the circle x^2-6 x+y^2-2 p y+17=0 , then find the value of (p₁^2+p₂^2 )

Step-by-step solution

The equation of given circle is aligned & S(x, y)=x^2+y^2-6 x-2 p y+17=0 (x-3)^2+(y-p)^2= (p^2-8 ) & S(0,0)=17 0 aligned (0,0) lies outside the circle. Equation of director circle of S=0 will be (x-3)^2+(y-p)^2=2 (p^2-8 ) . Tangents drawn from (0,0) to S=0 are perpendicular to each other (0,0) must lie on director circle. aligned & (0-3)^2+(0-p)^2=2 (p^2-8 ) & p^2=25 p= 5 aligned Hence p ₁^2+ p ₂^2=(5)^2+(-5)^2=25+25=50