Question

Find the coordinates of the point of intersection of the axis and the directrix of the parabola whose focus is (3, 3) and directrix is 3x – 4y = 2. Find also the length of the latus – rectum.

Answer 1

Given:

Focus = (3, 3)

Directrix = 3x – 4y = 2

Firstly let us find the slope of the directrix.

The slope of the line ax + by + c = 0 is –a/b

So, m₁ = -3/-4

= ¾

Let us assume the slope of axis is m₂.

m₁.m₂ = -1

¾ . m₂ = -1

m₂ = -4/3

We know that the equation of the line passing through the point (x₁, y₁) and having slope m is (y – y₁) = m(x – x₁)

y – 3 = -4/3 (x – 3)

3(y – 3) = – 4(x – 3)

3y – 9 = – 4x + 12

4x + 3y = 21

On solving the lines, the intersection point is (18/5, 11/5)

By using the formula to find the length is given as

∴The length of the latus rectum is 2.