Question

Find the equation of the parabola whose:
(i) focus is (3, 0) and the directrix is 3x + 4y = 1

(ii) focus is (1, 1) and the directrix is x + y + 1 = 0

(iii) focus is (0, 0) and the directrix is 2x – y – 1 = 0

(iv) focus is (2, 3) and the directrix is x – 4y + 1 = 0

Answer 1

(i) focus is (3, 0) and the directrix is 3x + 4y = 1

Given:

The focus S(3, 0) and directrix(M) 3x + 4y – 1 = 0.

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

The distance between two points (x₁, y₁) and (x₂, y₂) is given as:

And the perpendicular distance from the point (x₁, y₁) to the line ax + by + c = 0 is 

So by equating both, we get

Upon cross multiplication, we get

25x² + 25y² – 150x + 225 = 9x² + 16y² – 6x – 8y + 24xy + 1

16x² + 9y² – 24xy – 144x + 8y + 224 = 0

∴The equation of the parabola is 16x² + 9y² – 24xy – 144x + 8y + 224 = 0

(ii) focus is (1, 1) and the directrix is x + y + 1 = 0

Given:

The focus S(1, 1) and directrix(M) x + y + 1 = 0.

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

The distance between two points (x₁, y₁) and (x₂, y₂) is given as:

And the perpendicular distance from the point (x₁, y₁) to the line ax + by + c = 0 is 

So by equating both, we get

Upon cross multiplication, we get

2x² + 2y² – 4x – 4y + 4 = x² + y² + 2x + 2y + 2xy + 1

x² + y² + 2xy – 6x – 6y + 3 = 0

∴ The equation of the parabola is x² + y² + 2xy – 6x – 6y + 3 = 0

(iii) focus is (0, 0) and the directrix is 2x – y – 1 = 0

Given:

The focus S(0, 0) and directrix(M) 2x – y – 1 = 0.

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

The distance between two points (x₁, y₁) and (x₂, y₂) is given as:

And the perpendicular distance from the point (x₁, y₁) to the line ax + by + c = 0 is 

So by equating both, we get

Upon cross multiplication, we get

5x² + 5y² = 4x² + y² – 4x + 2y – 4xy + 1

x² + 4y² + 4xy + 4x – 2y – 1 = 0

∴ The equation of the parabola is x² + 4y² + 4xy + 4x – 2y – 1 = 0

(iv) focus is (2, 3) and the directrix is x – 4y + 1 = 0

Given:

The focus S(2, 3) and directrix(M) x – 4y + 3 = 0.

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

The distance between two points (x₁, y₁) and (x₂, y₂) is given as:

And the perpendicular distance from the point (x₁, y₁) to the line ax + by + c = 0 is 

So by equating both, we get

Upon cross multiplication, we get

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