# Focal Distance of a Point on the Ellipse

What is the focal distance of a point on the ellipse?

The sum of the focal distance of any point on an ellipse is constant and equal to the length of the major axis of the ellipse.

Let P (x, y) be any point on the ellipse $$\frac{x^{2}}{a^{2}}$$ + $$\frac{y^{2}}{b^{2}}$$ = 1.

Let MPM' be the perpendicular through P on directrices ZK and Z'K'. Now by definition we get,

SP = e  PM

⇒ SP = e ∙ NK

⇒ SP = e (CK - CN)

⇒ SP = e($$\frac{a}{e}$$ - x)

⇒ SP = a - ex ………………..…….. (i)

and

S'P = e PM'

⇒ S'P = e (NK')

⇒ S'P = e (CK' + CN)

⇒ S'P = e ($$\frac{a}{e}$$ + x)

⇒ S'P = a + ex ………………..…….. (ii)

Therefore, SP + S'P = a - ex + a + ex = 2a = major axis.

Hence, the sum of the focal distance of a point P (x, y) on the ellipse $$\frac{x^{2}}{a^{2}}$$ + $$\frac{y^{2}}{b^{2}}$$ = 1 is constant and equal to the length of the major axis (i.e., 2a) of the ellipse.

Note: This property leads to an alternative definition of ellipse as follows:

If a point moves on a plane in such a way that the sum of its distances from two fixed points on the plane is always a constant then the locus traced out by the moving point on the plane is called an ellipse and the two fixed points are the two foci of the ellipse.

Solved example to find the focal distance of any point on an ellipse:

Find the focal distance of a point on the ellipse 25x$$^{2}$$ + 9y$$^{2}$$ -150x – 90y + 225 = 0

Solution:

The given equation of the ellipse is 25x$$^{2}$$ + 9y$$^{2}$$ - 150x - 90y + 225 = 0.

From the above equation we get,

25x$$^{2}$$ - 150x + 9y$$^{2}$$ - 90y = - 225

⇒ 25(x$$^{2}$$ - 6x) + 9(y$$^{2}$$ - 10y) = -225

⇒ 25(x$$^{2}$$ - 6x + 9) + 9(y$$^{2}$$ - 10y + 25) = 225

⇒ 25(x - 3)$$^{2}$$ + 9(y - 5)$$^{2}$$ = 225

⇒ $$\frac{(x - 3)^{2}}{9}$$ + $$\frac{(y - 5)^{2}}{25}$$ = 1 ………………….. (i)

Now transfering the origin at (3, 5) without rotating the coordinate axes and denoting the new coordinates with respect to the new axes by x and y, we have

x = X + 3 and y = Y + 5 ………………….. (ii)

Using these relations, equation (i) reduces to

$$\frac{X^{2}}{3^{2}}$$ + $$\frac{Y^{2}}{5^{2}}$$ = 1 ……………………… (iii)

This is the form of $$\frac{X^{2}}{b^{2}}$$ + $$\frac{Y^{2}}{a^{2}}$$ = 1 (a$$^{2}$$ < b$$^{2}$$ ) where a = 5 and b = 3

Now, we get that a > b.

Hence, the equation$$\frac{X^{2}}{3^{2}}$$ + $$\frac{Y^{2}}{5^{2}}$$ = 1 represents an ellipse whose major axes along X and minor axes along Y axes.

Therefore, the focal distance of a point on the ellipse 25x$$^{2}$$ + 9y$$^{2}$$ - 150x - 90y + 225 = 0 is major axis = 2a = 2 5 = 10 units.

● The Ellipse

From Focal Distance of a Point on the Ellipse to HOME PAGE

Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.

## Recent Articles

1. ### Addition of Three 1-Digit Numbers | Add 3 Single Digit Numbers | Steps

Sep 19, 24 01:15 AM

To add three numbers, we add any two numbers first. Then, we add the third number to the sum of the first two numbers. For example, let us add the numbers 3, 4 and 5. We can write the numbers horizont…

2. ### Adding 1-Digit Number | Understand the Concept one Digit Number

Sep 18, 24 03:29 PM

Understand the concept of adding 1-digit number with the help of objects as well as numbers.

3. ### Addition of Numbers using Number Line | Addition Rules on Number Line

Sep 18, 24 02:47 PM

Addition of numbers using number line will help us to learn how a number line can be used for addition. Addition of numbers can be well understood with the help of the number line.

4. ### Counting Before, After and Between Numbers up to 10 | Number Counting

Sep 17, 24 01:47 AM

Counting before, after and between numbers up to 10 improves the child’s counting skills.