Question

Consider a family of circles passing through two fixed points a(3,7) and b(6,5) . Show that

Answer 1

Answer:

25

Step-by-step explanation:

Family of circles passing through A(3,7) and B(6,5) is

S+λP=0,  where S is the circle with AB as diameter and P is the equation of line AB.

⇒(x−3)(x−6)+(y−7)(y−5)+λ[y−7−  

6−3

5−7

​

(x−3)]=0

⇒(x−3)(x−6)+(y−7)(y−5)+λ(2x+3y−27)=0

⇒x  

2

+y  

2

+(2λ−9)x+(3λ−12)y+53−27λ=0

Common chord  is S  

1

​

−S  

2

​

=0.

⇒(2λ−5)x+(3λ−6)y+56−27λ=0

⇒(−5x−6y+56)+λ(2x+3y−27)=0

This chord is the intersection of −5x−6y+56=0 and 2x+3y−27=0.

Solving the above equations, we get (a,b)=(2,  

3

23

​

).

∴a+3b=25