Question

Consider a circle S:(x-11)^2+(y-17)^2= 1 and a fixed point P(5, 6). Two varlable polnt Q and R lle on circle and line y = x
respectively, then mlnlmum value of PR + QR, Is I then the value of lambda/10 is​

Answer 1

Given circle is x2+y2+8x+4y−5=0    ...(1)

Let the equation of the second circle be x2+y2+2gx+2fy+c=0

Since it passes through origin ∴c=0

So, the equation becomes x2+y2+2gx+2fy=0    ...(2)

The equation of common chord of (1) and (2) is

2(g−4)x+2(f−2)y+5=0   ...(3)

Since the line y=x touches the circle (2)

∴x2+x2+2gx+2fx=0 has equal roots

⇒f+g=0

∴ From (3), the equation of common chord is

2(g−4)x+2(−g−2)y+5=0⇒(−8x−4y+5)+g(2x−2y)=0

which passes through the point of intersection of 

8x+4y−5=0 and x=y 

i.e.  the point is (125,125)