Question

Circles are drawn with diameter being any focal chord of the parabola $y^2-4x-y-4=0$ will always touch the fixed line then its equation is

• A. $16x-33=0$
• B. $16x+33=0$
• C. $8x-33=0$
• D. $8x+33=0$

Answer 1

The correct option is B $16x+33=0$

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

$y^2-4x-y-4=0$

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

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

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

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 $X=-1$

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

$\Rightarrow x=-1-\frac{17}{16}=\frac{-16-17}{16}=\frac{-33}{16}$

$\therefore 16x+33=0$ is the equation of the directrix.