Circle

Circle I : x2+y2−6x−4y+9=0
Circle II : x2+y2+2x+2y−7=0
Point of contact : (α,β)
⇒(x−3)2+(y−2)2=4
Center =(3,2)
(x+1)2+(y+1)2=9
centre (−1,−1)
Slope of line (m)=‌

3

4

Equation of line y−2=‌

3

4

(x−3)
⇒3x−4y−1=0
The point of contact ( α,β ) lies on the line 3x−4y−1=0
⇒ Radii are r1=2 and r2=3
Ratio =2:3
Using section formula
‌α=‌

2(−1)+3(3)

2+3

=‌

7

5

,β=‌

2(−1)+3(2)

2+3

‌⇒β=‌

4

5

⇒7β=7×‌

4

5

=4⋅‌

7

5

‌∴7β=4α