# Ellipse Examples

Here you will learn some ellipse examples for better understanding of ellipse concepts.

Example 1 : Find the equation of ellipse whose foci are (2, 3), (-2, 3) and whose semi major axis is of length $$\sqrt{5}$$

Solution : Here S = (2, 3) & S’ is (-2, 3) and b = $$\sqrt{5}$$ $$\implies$$ SS’ = 4 = 2ae $$\implies$$ ae = 2

but $$b^2$$ = $$a^2(1-e^2)$$ $$\implies$$ 5 = $$a^2$$ – 4 $$\implies$$ a = 3

Hence the equation to major axis is y = 3.

Centre of ellipse is midpoint of SS’ i.e. (0, 3)

$$\therefore$$   Equation to ellipse is $$x^2\over a^2$$ + $${(y-3)}^2\over b^2$$ = 1 or $$x^2\over 9$$ + $${(y-3)}^2\over 5$$ = 1

Example 2 : For what value of k does the line y = x + k touches the ellipse $$9x^2 + 16y^2$$ = 144.

Solution : $$\because$$ Equation of ellipse is $$9x^2 + 16y^2$$ = 144 or $$x^2\over 16$$ + $${(y-3)}^2\over 9$$ = 1

comparing this with $$x^2\over a^2$$ + $$y^2\over b^2$$ = 1 then we get $$a^2$$ = 16 and $$b^2$$ = 9

and comparing the line y = x + k with y = mx + c   m = 1 and c = k

If the line y = x + k touches the ellipse $$9x^2 + 16y^2$$ = 144, then $$c^2$$ = $$a^2m^2 + b^2$$

$$\implies$$ $$k^2$$ = 16 $$\times$$ $$1^2$$ + 9 $$\implies$$ $$k^2$$ = 25

$$\therefore$$   k = $$\pm$$5

Example 3 : Find the equation of the tangents to the ellipse $$3x^2+4y^2$$ = 12 which are perpendicular to the line y + 2x = 4

Solution : Let m be the slope of the tangent, since the tangent is perpendicular to the line y + 2x = 4

$$\therefore$$   mx – 2 = -1 $$\implies$$ m = $$1\over 2$$

Since $$3x^2+4y^2$$ = 12 or $$x^2\over 4$$ + $$y^2\over 3$$ = 1

Comparing this with $$x^2\over a^2$$ + $$y^2\over b^2$$ = 1

$$\therefore$$   $$a^2$$ = 4 and $$b^2$$ = 3

So the equation of the tangent are y = $$1\over 2$$x $$\pm$$ $$\sqrt{4\times {1\over 4} + 3}$$

$$\implies$$ y = $$1\over 2$$x $$\pm$$ 2 or x – 2y $$\pm$$ 4 = 0

Practice these given ellipse examples to test your knowledge on concepts of ellipse.