Find the equation of the parabola given vertex and directrix? The vertex of parabola is $V(3, 1)$ and the directrix is $4x + 3y = 5$. I can't figure out the focus by using the axis of symmetry which is $3x - 4y -5 =0$.

Here is a straightforward way to find the unique focal point. From the directrix and the symmetry lines $$ 4x+3y=5, \>\>\>\>\>3x-4y=5$$

their intersection is $(\frac75, -\frac15)$. Since the vertex is the midpoint between the focus $(a,b)$ and the intersection point,

$$3=\frac{a+\frac75}{2},\>\>\>\>\>1=\frac{b-\frac15}{2}$$

Then, solve for the focus $(a, b)$.