14) Find the equation of circle passing through thepoints (1, -2) and

1.	14) Find the equation of circle passing through thepoints (1, -2) and (4,-3) and whose centre lies onthe line 3x +2y 7.
Answer» equation of circle with(1,−2)(1,−2)and as(4,−3)(4,−3)diameter, (x−1)(x−4)+y+2)(y+3)=0(1)(1)(x−1)(x−4)+y+2)(y+3)=0 Equation of line joining(1,−2)(1,−2)and as(4,−3)(4,−3) (4–1)(y+2)−(−3+2)(x−1)=0⟹(4–1)(y+2)−(−3+2)(x−1)=0⟹ 3y+x+5=0(2)(2)3y+x+5=0 equation of family of circles passing through the 2 points is (x−1)(x−4)+(y+2)(y+3)−2K(3y+x+5)=0⟹(x−1)(x−4)+(y+2)(y+3)−2K(3y+x+5)=0⟹ x2+y2–2(K+52)x−2(3K−52)y+10–10K=0(3)(3)x2+y2–2(K+52)x−2(3K−52)y+10–10K=0 The center is(K+52,3K−52)(K+52,3K−52) This should lie on3x+2y=7⟹3x+2y=7⟹ 3(K+52)+2(3K−52)=7⟹2K=13(K+52)+2(3K−52)=7⟹2K=1 (3) now becomes x2+y2–6x+2y+5=0x2+y2–6x+2y+5=0

14) Find the equation of circle passing through thepoints (1, -2) and (4,-3) and whose centre lies onthe line 3x +2y 7.

Answer»

equation of circle with(1,−2)(1,−2)and as(4,−3)(4,−3)diameter,

(x−1)(x−4)+y+2)(y+3)=0(1)(1)(x−1)(x−4)+y+2)(y+3)=0

Equation of line joining(1,−2)(1,−2)and as(4,−3)(4,−3)

(4–1)(y+2)−(−3+2)(x−1)=0⟹(4–1)(y+2)−(−3+2)(x−1)=0⟹

3y+x+5=0(2)(2)3y+x+5=0

equation of family of circles passing through the 2 points is

(x−1)(x−4)+(y+2)(y+3)−2K(3y+x+5)=0⟹(x−1)(x−4)+(y+2)(y+3)−2K(3y+x+5)=0⟹

x2+y2–2(K+52)x−2(3K−52)y+10–10K=0(3)(3)x2+y2–2(K+52)x−2(3K−52)y+10–10K=0

The center is(K+52,3K−52)(K+52,3K−52)

This should lie on3x+2y=7⟹3x+2y=7⟹

3(K+52)+2(3K−52)=7⟹2K=13(K+52)+2(3K−52)=7⟹2K=1

(3) now becomes

x2+y2–6x+2y+5=0x2+y2–6x+2y+5=0