Question

Find the equation of the parabola satisfying the following conditions;

1. Focus(6, 0); directrix x = – 6

2. Vertex (0, 0); Focus (3, 0)

3. Vertex (0, 0) passing through (2, 3) and axis along x-axis

Answer 1

1. Since the focus (6, 0) lies on the x-axis, therefore x-axis is the axis of parabola.

Also the directrix is x = – 6, ie; x = – a And focus (6, 0), ie; (a, 0)

Therefore the equation of the parabola is

y² = 4ax ⇒ y² = 24x.

2. The vertex of the parabola is at (0, 0) and focus is at (3, 0). 

Then axis of parabola is along x-axis. So the parabola is of the form y² = 4ax . 

The equation of the parabola is y² = 12x.

3. The vertex of the parabola is at (0, 0) and the axis is along x-axis. 

So the equation of parabola is of the torn y² = 4ax .

Since the parabola passes through point (2, 3)

Therefore, 3² = 4a × 2 ⇒ a = \(\frac{9}{8}\)

The required equation of the parabola is

y² = 4 × \(\frac{9}{8}\) x ⇒ y² = \(\frac{9}{2}\)x.