## Question

To find the equation of an ellipse from the definition that ellipse is the locus of a point which moves such that the sum of its distances from two fixed points is constant with the fixed points as foci.

### Solution

Let two fixed points be *S*(*ae*, 0) and *S*’(–*ae*, 0). Let *P*(*x*, *y*) be a moving point such that

or,

Equation (i) can be written as

From (2) and (3),

Adding (iv) and (v) then,

On squaring then

#### SIMILAR QUESTIONS

Find the length of major and minor axes the coordinates of foci and vertices and, the eccentricity of the ellipse 3*x*^{2} + 2*y ^{2}* = 6. Also, find the equation of the directrices.

Find the equation of an ellipse whose focus is (–1, 1), eccentricity is and the directrix is *x* – *y* + 3 = 0.

If the angle between the straight lines joining foci and the ends of the minor axis of the ellipse is 90^{o}. Find its eccentricity.

Find the equation of the ellipse referred to its centre whose minor axis is equal to the distance between the foci and whose latus rectum is 10.

The extremities of a line segment of length *l* move in two fixed perpendicular straights lines. Find the locus of that point which divides this line segment in ratio 1 : 2.

Find the lengths and equations of the focal radii drawn from the point on the ellipse 25*x*^{2} + 16*y*^{2} = 1600.

For what value of λ dose the line *y* = *x* + λ touches the ellipse

9*x*^{2} + 16*y*^{2} = 144.

Find the equations of the tangents to the ellipse which are perpendicular to the line *y* + 2*x* = 4.

Find the locus of the foot of the perpendicular drawn from centre upon any tangent to the ellipse .