Question

The equation of the parabola whose focus is the point (2, 3) and directrix is the line x – 4y + 3 = 0, is __________.

Answer 1

Let P(x, y) be any point an parabola,
Given focus is (2, 3)
∴ Distance between focus and P is, $\sqrt{{\left(x-2\right)}^2+{\left(x-3\right)}^2}$ ...(1)
Also directrix is given by x − 4y + 3 = 0
∴ The perpendicular distance from the point P to the line x − 4y + 3 = 0 is $\frac{\left|x-4y+3\right|}{\sqrt{1+16}}$
$\therefore \sqrt{{\left(x-2\right)}^2+{\left(y-3\right)}^2}-\frac{\left|x-4y+3\right|}{\sqrt{1+16}}$[since distance of any point an parabola plane focus is same as distance between point and directrix]

By squaring both sides,
$$\begin{gathered} \therefore {\left(x-2\right)}^2+{\left(y-3\right)}^2=\frac{{\left|x-4y+3\right|}^2}{17} \\ \therefore x^2-4x+4+y^2-6y+9=\frac{\left[{\left(x-4y\right)}^2+9+6x-24y\right]}{17} \\ \mathrm{i}.\mathrm{e}.17\left(x^2-4x-6y+y^2+13\right)=x^2+16y^2-8xy+9+6x-24y \\ \mathrm{i}.\mathrm{e}.17x^2+17y^2-68x-102y+221=x^2+16y^2+6x-24y+9-8xy \\ \mathrm{i}.\mathrm{e}.16x^2+y^2+8xy-74x-78y+212=0 \end{gathered}$$
is the required equation of parabola.