AP EAMCET Engineering 21 Sep 2020 Shift 2 Paper

(b) Equation of circle circumscribing the triangle formed by the straight lines, x+y=6,2x+y=4 and x+2y=5 is
‌(x+y−6)(2x+y−4)+λ(2x+y−4)(x+2y−5)‌+µ(x+2y−5)(x+y−6)=0
∵ For a circle the coefficients of terms x2 and y2 must be equal and non-zero and coefficient of xy must be zero.
∴ Coefficient of x2= coefficient of y2
⇒ 2+2 λ+µ=1+2 λ+2 µ ⇒ µ=1
and coefficient of xy=0 ⇒1+2+4λ+λ+µ+2µ=0⇒λ=−6∕5.
So, equation of required circle is
(x+y−6)(2x+y−4)−‌

6

5

(2x+y−4)(x+2y−5)+(x+2y−5)(x+y−6)=0
⇒5(x+y−6)(3x+3y−9)−6(2x+y−4)(x+2y−5)=0
⇒5(x+y−6)(x+y−3)−2(2x+y−4)(x+2y−5)=0
⇒5(x2+xy−3x+xy+y2−3y−6x−6y+18)−2(2x2+4xy−10x+xy+2y2−5y−4x−8y+20)
⇒x2+y2−45x−45y+90+28x+26y−40=0
⇒x2+y2−17x−19y+50=0