Question

From a variable point $P,$ normal is drawn to the hyperbola $xy=16.$ If sum of slopes of normal is equal to the sum of ordinates and abscissas of feet of normal, then the locus of point $P$ is the curve $C.$ Which of the following is (are) CORRECT ?

• A. $C$ is a parabola of length of latus rectum $16$ units
• B. $C$ is an ellipse of length of latus rectum $16$ units
• C. $C$ is a hyperbola of length of latus rectum $16$ units
• D. Focus of $C$ is at $(8,0)$

Answer 1

Answer: (A), (D)

Focus of $C$ is at $(8,0)$

Let $P\equiv (\alpha ,\beta )$ and foot of normal $\equiv (4t,\frac{4}{t})$
So, $\frac{\beta -\frac{4}{t}}{\alpha -4t}=t^2$
$\Rightarrow 4t^4-\alpha t^3+t\beta -4=0$

Let $t_1,t_2,t_3$ and $t_4$ be the roots.
Then $\sum 4t_1+\sum \frac{4}{t_1}=\sum t_1^2$
$\Rightarrow \alpha +\beta =\frac{{\alpha }^2}{16}$
$\therefore$ Locus of $P$ is $x^2=16(x+y)$
$\Rightarrow {(x-8)}^2=16(y+4)$
$4a=16\Rightarrow a=4$
Length of latus rectum $=|4a|=16$

Finding focus of $C:$
$x-8=0\Rightarrow x=8$
and $y+4=4\Rightarrow y=0$
$\Rightarrow$ Focus of $C$ is $(8,0)$