Question

Let $P(x,y)$ be a variable point such that $|\sqrt{(x-2{)}^2+(y-3{)}^2}-\sqrt{(x-6{)}^2+(y-6{)}^2}|=4$, then

• A. The locus of $P$ is a hyperbola with eccentricity $\frac{5}{4}$
• B. The locus of $P$ is a hyperbola with eccentricity $\frac{5}{3}$
• C. The locus of the intersection of perpendicular tangents to the locus of $P$ is ${(x-4)}^2+{(y-\frac{9}{2})}^2=\frac{5}{4}$
• D. The locus of the intersection of perpendicular tangents to the locus of $P$ is ${(x-4)}^2+{(y-\frac{9}{2})}^2=\frac{7}{4}$

Answer 1

Answer: (A), (D)

The correct options are
A The locus of $P$ is a hyperbola with eccentricity $\frac{5}{4}$
D The locus of the intersection of perpendicular tangents to the locus of $P$ is ${(x-4)}^2+{(y-\frac{9}{2})}^2=\frac{7}{4}$

$|SP-S^{\prime }P|=2a$ represents a hyperbola with $S$ & $S^{\prime }$ as focii and $2a$ as major axis.
Comparing with the given expression
$S\equiv (2,3)$, $S^{\prime }\equiv (6,6)$, $2a=4$
$S{S}^{\prime }=2ae$
$5=4e$
$e=\frac{5}{4}$
The locus of the intersection of perpendicular tangents to the hyperbola is its director circle
$a=2$, $e=\frac{5}{4}$
$\Rightarrow b=\frac{3}{2}$
$\Rightarrow$ director circle : ${(x-4)}^2+{(y-\frac{9}{2})}^2=\frac{7}{4}$