The equation of the parabola whose vertex is `(a ,0)` and the directrix has the equation `x+y=3a` is a.`x^2+y^2+2x y+6a x+10 a y+7a^2=0` b.`x^2-2x y+y^2-6a x+10 a y+7a^2=0` c.`x^2-2x y+y^2+6a x+10 a y+7a^2=0` d.None of these

Explanation

Equation of axis is given by, `x−y=c`, since the directrix and axis of the parabola are perpendicular to each other.

It also passes through vertex `(a,0)⇒c=a`

Now solving directrix and axis to get foot of directrix. `x=2a,y=a`

We know vertex is mid point of foot of directrix and focus. `∴`
focus is `S(0,−a)`

using definition of parabola,
`PS ^2
=PM^2
...