Question

Find the equation of the ellipse whose centre is (-2, 3) and whose semi – axis are 3 and 2 when the major axis is (i) parallel to x – axis (ii) parallel to the y – axis.

Answer 1

Given:

Centre = (-2, 3)

Semi – axis are 3 and 2

(i) When major axis is parallel to x-axis

Now let us find the equation to the ellipse.

We know that the equation of the ellipse with centre (p, q) is given by 

Since major axis is parallel to x – axis

So, a = 3 and b = 2.

a² = 9

b² = 4

So the equation of the ellipse can be given as

4(x² + 4x + 4) + 9(y² – 6y + 9) = 36

4x² + 16x + 16 + 9y² – 54y + 81 = 36

4x² + 9y² + 16x – 54y + 61 = 0

∴ The equation of the ellipse is 4x² + 9y² + 16x – 54y + 61 = 0.

(ii) When major axis is parallel to y-axis

Now let us find the equation to the ellipse.

We know that the equation of the ellipse with centre (p, q) is given by 

Since major axis is parallel to y – axis

So, a = 2 and b = 3.

a² = 4

b² = 9

So the equation of the ellipse can be given as

9(x² + 4x + 4) + 4(y² – 6y + 9) = 36

9x² + 36x + 36 + 4y² – 24y + 36 = 36

9x² + 4y² + 36x – 24y + 36 = 0

∴ The equation of the ellipse is 9x² + 4y² + 36x – 24y + 36 = 0.