Question

Let $A(2,3)$ and $B(-4,5)$ be two fixed points, a point $P$ move in such a way that $△PAB=12$ sq units, them its locus is

• A. $x^2+6xy+9y^2+22x+66y-23=0$
• B. $x^2+6xy+9y^2+22x+66y+23=0$
• C. $x^2+y^2+2x+4y+1=0$
• D. $x^2+6xy+9y^2-22x-66y-23=0$

Answer 1

Answer: (D)

$x^2+6xy+9y^2-22x-66y-23=0$

Let, $P=(x,y)$

$A=(2,3)$ and $B=(-4,5)$

$\because Ar.\left(△PAB\right)=12$ sq. units

$⟹\frac{1}{2}|x(3-5)+2(5-y)-4(y-3)|=12$

$⟹|-2x+10-2y-4y+12|=24$

$⟹|-2x-6y+22|=24$

$⟹2x+6y-22=\pm 24$

$⟹x+3y-11=\pm 12$

$⟹x+3y-23=0$ or $x+3y+1=0$

Hence, the locus is

$(x+3y-23)(x+3y+1)=0$

$⟹x^2+6xy+9y^2-22x-66y-23=0$ (Option D)