Question

Find the equation of the parabola whose focus is $(1,-1)$ and vertex is $(2,1)$.

Answer 1

Vertex form of parabola whose vertex is $(h,k)$.

${(y-k)}^2=4a(x-h)$

where as,

$a=$ distance between vertex and focus of parabola.

$\because$ vertex is given, i.e., $(2,1)$,

$\therefore$ Equation of parabola is-

${(y-1)}^2=4a(x-2)$

As the focus is given, i.e., $(1,-1)$,

$\therefore a=\sqrt{{(2-1)}^2+{(1-(-1))}^2}$

$\Rightarrow a=\sqrt{1^2+2^2}$

$\Rightarrow a=\sqrt{1+4}=\sqrt{5}$

$\therefore$ Equation of parabola is-

${(y-1)}^2=4\times \sqrt{5}\times (x-2)$

${(y-1)}^2=4\sqrt{5}(x-2)$