Question

Let P be the parabola with focus (0,4) and directrix y=x.Write an equation whose graph is a parabola with a vertical directrix that is congruent to P.

Answer 1

Answer:

y²=4√2.x

Step-by-step explanation:

The focus is at (0,4) and directrix is y=x or x-y =0, for a parabola P.

The distance between the focus and the directrix of the parabola P is

$\frac{ |0-4| }{\sqrt{(1)^{2}+(-1)^{2} } }$=$\frac{4}{\sqrt{2} }$

{Since the perpendicular distance of a point (x1, y1) from the straight line ax+by+c =0 is given by $\frac{ |ax1+by1+c| }{\sqrt{a^{2}+b^{2} } }$ }

Let us assume that the equation of the parabola which is congruent with parabola P is y²=4ax

{Since the parabola has vertical directrix}

Hence, the distance between focus and the directrix is 2a = $\frac{4}{\sqrt{2} }$, {Two parabolas are congruent when the distances between their focus and the directrix are same}

⇒ a=√2

Therefore, the equation of the parabola is y²=4√2.x (Answer)