find the focus and equation of parabola whose vertex is (6,-3) and diretrix is 3x - 5y + 1 = 0.
Equation of directrix=3x - 5y + 1 = 0--------(i)
Let the coordinate of focus = (a,b)
the axis of parabola is perpendicular to directrix.
So the equation of axis of parabola may be taken as x+y+k=0
it passes through (6,-3)
⇒ the equation of axis of parabola is x+y-3=0---------------(ii)
Now, for the point of intersection of directrix and axis of parabola
On solving (i) and (ii), we get,
put the value of y in (i), we get,
Thus, the point of intersection of directrix and axis of parabola
Vertex A (6,-3) is the mid-point of focus and
Coordinate of focus =
which is the equation of parabola