Question

Find the equation of the parabola whose : 

Focus is (2, 3) and the directrix is x - 4y + 1 = 0

Answer 1

Given that,

We need to find the equation of the parabola whose focus is S(2, 3) and directrix(M) is x - 4y + 3 = 0.

Let us assume P(x, y) be any point on the parabola. 

We know that,

The point on the parabola is equidistant from focus and directrix.

We know that,

The distance between two points (x₁, y₁) and (x₂, y₂) is \(\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}\).

We know that,

The perpendicular distance from a point (x₁, y₁) to the line ax + by + c = 0 is \(\frac{|ax_1+by_1+c|}{\sqrt{a^2+b^2}}\).

⇒ SP = PM 

⇒ SP² = PM²

⇒ 17x² + 17y² - 68x - 102y + 221 = x² + 16y² + 6x - 24y - 8xy + 9 

⇒ 16x² + y² + 8xy - 74x - 78y + 212 = 0 

∴The equation of the parabola is 16x² + y² + 8xy - 74x - 78y + 212 = 0.