Let A , (2, 3) and B , (-4, 5) are two fixed points. A point P mo… - Vidyadip

MCQ

Let $A\, (2, 3)$ and $B\, (-4, 5)$ are two fixed points. $A$ point $P$ moves in such a way that $\Delta PAB = 12 \,sq. units$, then its locus is :-

A
$x^2 + 6xy + 9y^2 + 22 x + 66 y -23 = 0$
B
$x^2 + 6xy + 9y^2 + 22 x + 66 y + 23 = 0$
✓
$x^2 + 6xy + 9y^2 -22 x -66 y -23 = 0$
D
none of these

✓

Answer

Correct option: C.

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

c
Let, $P=(x, y)$

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

$\because A r .(\triangle P A B)=12$ sq. units

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

$\Longrightarrow|-2 x+10-2 y-4 y+12|=24$

$\Longrightarrow|-2 x-6 y+22|=24$

$\Longrightarrow 2 x+6 y-22=\pm 24$

$\Rightarrow x+3 y-11=\pm 12$

$\Longrightarrow x+3 y-23=0$ or $x+3 y+1=0$

Hence, the locus is $(x+3 y-23)(x+3 y+1)=0$

$\Longrightarrow x^{2}+6 x y+9 y^{2}-22 x-66 y-23=0$

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