Question

Find the vertex, focus, axis, directrix and lotus – rectum of the following parabolas
(i) y² = 8x

(ii) 4x² + y = 0

(iii) y² – 4y – 3x + 1 = 0

(iv) y² – 4y + 4x = 0

(v) y² + 4x + 4y – 3 = 0

Answer 1

(i) y² = 8x

Given:

Parabola = y² = 8x

Now by comparing with the actual parabola y² = 4ax

Then,

4a = 8

a = 8/4 = 2

So, the vertex is (0, 0)

The focus is (a, 0) = (2, 0)

The equation of the axis is y = 0.

The equation of the directrix is x = – a i.e., x = – 2

The length of the latus rectum is 4a = 8.

(ii) 4x² + y = 0

Given:

Parabola => 4x² + y = 0

Now by comparing with the actual parabola y² = 4ax

Then,

4a = 1/4

a = 1/(4 × 4)

= 1/16

So, the vertex is (0, 0)

The focus is = (0, -1/16)

The equation of the axis is x = 0.

The equation of the directrix is y = 1/16

The length of the latus rectum is 4a = 1/4

(iii) y² – 4y – 3x + 1 = 0

Given:

Parabola y² – 4y – 3x + 1 = 0

 y² – 4y = 3x – 1

y² – 4y + 4 = 3x + 3

(y – 2)² = 3(x + 1)

Now by comparing with the actual parabola y² = 4ax

Then,

4b = 3

b = 3/4

So, the vertex is (-1, 2)

The focus is = (3/4 – 1, 2) = (-1/4, 2)

The equation of the axis is y – 2 = 0.

The equation of the directrix is (x – c) = -b

(x – (-1)) = -3/4

x = -1 – 3/4

= -7/4

The length of the latus rectum is 4b = 3

(iv) y² – 4y + 4x = 0

Given:

Parabola y² – 4y + 4x = 0

y² – 4y = – 4x

y² – 4y + 4 = – 4x + 4

(y – 2)² = – 4(x – 1)

Now by comparing with the actual parabola y² = 4ax => (y – a)² = – 4b(x – c)

Then,

4b = 4

b = 1

So, the vertex is (c, a) = (1, 2)

The focus is (b + c, a) = (1-1, 2) = (0, 2)

The equation of the axis is y – a = 0 i.e., y – 2 = 0

The equation of the directrix is x – c = b

x – 1 = 1

x = 1 + 1

= 2

Length of latus rectum is 4b = 4

(v) y² + 4x + 4y – 3 = 0

Given:

The parabola y² + 4x + 4y – 3 = 0

y² + 4y = – 4x + 3

y² + 4y + 4 = – 4x + 7

(y + 2)² = – 4(x – 7/4)

Now by comparing with the actual parabola y² = 4ax => (y – a)² = – 4b(x – c)

Then,

4b = 4

b = 4/4 = 1

So, The vertex is (c, a) = (7/4, -2)

The focus is (- b + c, a) = (-1 + 7/4, -2) = (3/4, -2)

The equation of the axis is y – a = 0 i.e., y + 2 = 0

The equation of the directrix is x – c = b

x – 7/4 = 1

x = 1 + 7/4

= 11/4

Length of latus rectum is 4b = 4.