Question

The focus and directrix of a parabola are ( $1,-1$ ) and $x+y+3=0.$ Its vertex is

• A. $(\frac{-3}{4},\frac{7}{4})$
• B. $(\frac{1}{2},\frac{-7}{4})$
• C. $(\frac{1}{4},\frac{-7}{4})$
• D. $(\frac{1}{2},\frac{5}{2})$

Answer 1

The correct option is A $(\frac{-3}{4},\frac{7}{4})$

Equation of directrix $=x+y+3=0$

Equation of line through focus $=x-y+\lambda =0$

It passes through $(1,-1)$

$\therefore x-y+\lambda =0$

$\Rightarrow 1-(-1)+\lambda =0\Rightarrow \lambda =-2$

$\therefore$ equation is $x-y-2=0\rightarrow (1)$

Solving $\left(1\right)$ and $\left(2\right)$ we get point of intersection as

$x=\frac{-1}{2};y=\frac{-5}{2}$

$\therefore$ Vertex is $\frac{\frac{-1}{2}+1}{2},\frac{\frac{-5}{2}-1}{2}=\frac{-3}{4},\frac{-7}{4}$