Solving Ellipse

A plane curve formed by the intersection of a cone and a plane (i.e., a closed curve) is known as an Ellipse. If the cutting plane is normal to the axis, then a special case of ellipse is obtained called circle. An Ellipse is a closed curve and it is a bounded case of a conic section.

I like to share this Major Axis of an Ellipse with you all through my article.

Solving an Ellipse:

Let us see about the solving ellipse,

If P is a point on ellipse, then the sum of the distances from P to the two focus is a constant.

The equation for ellipse with normal major axis.

The derivative for the horizontal major axis case is the same.

Let us consider the top of  an ellipse, first.

The sum of the distances from P to the two focus,

(b - c) + (b + c) = 2b (top of  an ellipse)

squaring the above equation,

= 4b2

now consider the R.H.S. of  ellipse.

Thus, the distance from each focus to the R.H.S is the same.

Squaring the sum, implies

(2d)2 = 4d2

by Pythagoras theorem,

d2 = a2 + c2

Sub. And equate both the equations

4b2 = 4(a2 + c2)

simplifying, implies

b2 = a2 + c2

The equation of an ellipse is solving with center as its origin, can be given by

(x^2)/(a^2) + (y^2)/(b^2) = 1

with a > b > 0.

Here the major axis is 2a and the minor axis is 2b.

The two foci are given by (±c , 0),(0 , ±c) where c2 = a2 - b2.

Hence, solving ellipse proved.

Example Problem - Solving Ellipse:

Find  x and y intercepts of the following ellipse.

9x2 + 4y2 = 36

Solution:

To solving the above equation, write the ellipse  equation in  general form

(9x^2) / 36 + (4y^2) / 36 = 1

by simplification,

x^2 / 4 + y^2 / 9 = 1

=>  x^2 / 2^2 + y^2 / 3^2 = 1

from the above ellipse equation, the x and y-intercepts are obtained. i.e., a = 3 and b = 2.

sub. y=0 =>  x^2 / 2^2 = 1

by Solving x.

x2 = 22

x = ± 2

sub. x=0 => y^2 / 3^2 = 1

by Solving y.

y2 = 32

y = ± 3.