Question

Circle drawn with a focal chord of the parabola $y^2=4x+y+4$ as diameter will always touch the line $16x+k=0,$ where $k$ equals to :

• A. $8$
• B. $16$
• C. $33$
• D. $36$

Answer 1

The correct option is C $33$

Whenever a circle is drawn with diameter being focal chord of the parabola then, the circle touch the directrix.

$y^2=4x+y+4$

$\Rightarrow$ $y^2-y=4x+4$

$\Rightarrow$ $y^2-y+\frac{1}{4}=4x+4+\frac{1}{4}$

$\Rightarrow$ ${(y-\frac{1}{2})}^2=4(x+\frac{17}{6})$

Let $Y=y-\frac{1}{2}$ and $X=x+\frac{17}{6}$

We have,

$Y^2=4X$

$\therefore$ $a=1$

$\therefore$ Equation of directrix is $X=-1$

$\Rightarrow$ $x+\frac{17}{16}=-1$

$\Rightarrow$ $x=-1-\frac{17}{16}$

$\Rightarrow$ $x=\frac{-16-17}{16}$

$\Rightarrow$ $x=\frac{-33}{16}$

$\Rightarrow$ $16x+33=0$

The equation of directrix $16x+33=0$ comparing it with $16x+k=0$ we get,

$\Rightarrow$ $k=33$