Chapter 11.5 - Simplest Equation of Ellipse

In the previous section, we saw the basic properties of an ellipse. In this section, we will see equation of an ellipse.

• To write the equation of an ellipse, we must first place it on the Cartesian plane.
• The equation will be in the simplest form when the following three conditions are satisfied:
    ♦ The center of the ellipse is at the origin O.
    ♦ The major axis of the ellipse lies along the x-axis.
    ♦ The minor axis of the ellipse lies along the y-axis.
• This is shown in fig.11.34 below:

Fig.11.34

• Based on fig.11.34, we can derive the equation in 6 steps:
1. Let P(x,y) be any point on the ellipse.
2. We know that, F₁ is at a distance of ‘c’ from O.
• So the coordinates of F₁ will be (-c,0)   
3. We know that, F₂ is at a distance of ‘c’ from O.
• So the coordinates of F₂ will be (c,0)
4. Now we have three points and their coordinates:
P(x,y), F₁(-c,0), F₂(c,0)
• Using the distance formula, we can write some distances:

• First we write the distance PF₁:
$\begin{array}{ll} {}&{PF_1} & {~=~}& {\sqrt{(x~-~ -c)^2~+~(y - 0)^2}} &{} \\ {}&{} & {~=~}& {\sqrt{(x + c)^2~+~y^2}} &{} \\ \end{array}$ 

• Next we write the distance PF₂:
$\begin{array}{ll} {}&{PF_2} & {~=~}& {\sqrt{(x~-~ c)^2~+~(y - 0)^2}} &{} \\ {}&{} & {~=~}& {\sqrt{(x - c)^2~+~y^2}} &{} \\ \end{array}$

• Sum of the above two distances is: $\sqrt{(x + c)^2~+~y^2}~+~\sqrt{(x - c)^2~+~y^2}$

5. We know that, B is at a distance of 'a' from O. So the coordinates of B will be (a,0).
• Now we have three points and their coordinates:
B(a,0), F₁(-c,0), F₂(c,0)
• Using the distance formula, we can write some distances:

• First we write the distance BF₁:
$\begin{array}{ll} {}&{BF_1} & {~=~}& {\sqrt{(a~-~ -c)^2~+~(0 - 0)^2}} &{} \\ {}&{} & {~=~}& {\sqrt{(a + c)^2}} &{} \\ {}&{} & {~=~}& {a + c} &{} \\ \end{array}$ 

• Next we write the distance BF₂:
$\begin{array}{ll} {}&{BF_2} & {~=~}& {\sqrt{(a~-~ c)^2~+~(0 - 0)^2}} &{} \\ {}&{} & {~=~}& {\sqrt{(a - c)^2}} &{} \\ {}&{} & {~=~}& {a - c} &{} \\ \end{array}$

• Sum of the above two distances is: (a+c) + (a-c) = 2a

6. Both P and B are points on the same ellipse. So the sum of the distances must be equal.
Equating the results in (4) and (5), we get:

$\begin{array}{ll} {}&{\sqrt{(x + c)^2~+~y^2}~+~\sqrt{(x - c)^2~+~y^2}} & {~=~}& {2a} &{} \\ {\Rightarrow}&{\sqrt{(x + c)^2~+~y^2}} & {~=~}& {2a~-~\sqrt{(x - c)^2~+~y^2}} &{} \\ {\Rightarrow}&{(x + c)^2~+~y^2} & {~=~}& {4a^2~-~4a \sqrt{(x - c)^2~+~y^2}~+~(x - c)^2~+~y^2~~ \color {green} {\text{- - - (I)}}} &{} \\ {\Rightarrow}&{x^2 + 2xc + c^2 + y^2} & {~=~}& {4a^2~-~4a \sqrt{(x - c)^2~+~y^2}~+~x^2 - 2xc + c^2~+~y^2} &{} \\ {\Rightarrow}&{2xc} & {~=~}& {4a^2~-~4a \sqrt{(x - c)^2~+~y^2}- 2xc} &{} \\ {\Rightarrow}&{4xc} & {~=~}& {4a^2~-~4a \sqrt{(x - c)^2~+~y^2}} &{} \\ {\Rightarrow}&{xc} & {~=~}& {a^2~-~a \sqrt{(x - c)^2~+~y^2}~~ \color {green} {\text{- - - (II)}}} &{} \\ {\Rightarrow}&{\frac{xc}{a}} & {~=~}& {a~-~\sqrt{(x - c)^2~+~y^2}} &{} \\ {\Rightarrow}&{\sqrt{(x - c)^2~+~y^2}} & {~=~}& {a~-~\frac{xc}{a}~~ \color {green} {\text{- - - (III)}}} &{} \\ {\Rightarrow}&{(x - c)^2~+~y^2} & {~=~}& {a^2~-~\frac{2axc}{a}~+~\frac{x^2 c^2}{a^2}} &{} \\ {\Rightarrow}&{(x - c)^2~+~y^2} & {~=~}& {a^2~-~2cx~+~\frac{x^2 c^2}{a^2}} &{} \\ {\Rightarrow}&{x^2 - 2cx + c^2~+~y^2} & {~=~}& {a^2~-~2cx~+~\frac{x^2 c^2}{a^2}} &{} \\ {\Rightarrow}&{x^2 + c^2~+~y^2} & {~=~}& {a^2~+~\frac{x^2 c^2}{a^2}} &{} \\ {\Rightarrow}&{x^2 ~+~y^2~-~\frac{x^2 c^2}{a^2}} & {~=~}& {a^2 - c^2} &{} \\ {\Rightarrow}&{x^2 \left(1~-~\frac{c^2}{a^2} \right)~+~y^2} & {~=~}& {a^2 - c^2} &{} \\ {\Rightarrow}&{x^2 \left(\frac{a^2~-~c^2}{a^2} \right)~+~y^2} & {~=~}& {a^2 - c^2~~ \color {green} {\text{- - - (IV)}}} &{} \\ {\Rightarrow}&{x^2 \left(\frac{b^2}{a^2} \right)~+~y^2} & {~=~}& {b^2~~ \color {green} {\text{- - - (V)}}} &{} \\ {\Rightarrow}&{\frac{x^2}{a^2}~+~\frac{y^2}{b^2}} & {~=~}& {1} &{} \\ \end{array}$

◼ Remarks:
• Line marked as (I):
In this line, we square both sides.
• Line marked as (II):
In this line, we divide both sides by 4.
• Line marked as (III):
In this line, we square both sides.
• Line marked as (IV):
In this line, write b² in the place of a² - c².
• Line marked as (V):
In this line, we divide both sides by b².

---

Using the above 6 steps, we derived an equation. Now we will prove the converse. It can be written in 9 steps:

1. We derived an equation: $\frac{x^2}{a^2}~+~\frac{y^2}{b^2}~=~1$
2. To prove the converse, we assume a point P.
• Let P(x,y) be any point on the ellipse.
• Distance of P from F₁ can be written as:

$\begin{array}{ll} {}&{PF_1} & {~=~}& {\sqrt{(x~-~ -c)^2~+~(y - 0)^2}} &{} \\ {}&{} & {~=~}& {\sqrt{(x + c)^2~+~y^2}} &{} \\ \end{array}$

3. But based on the equation written in (1), we can write:

$\begin{array}{ll} {}&{\frac{y^2}{b^2}} & {~=~}& {1-\frac{x^2}{a^2}} &{} \\ {\Rightarrow}&{y^2} & {~=~}& {b^2\left(1-\frac{x^2}{a^2} \right)} &{} \\ \end{array}$

4. Substituting the above result in (2), we get:

$\begin{array}{ll} {}&{PF_1} & {~=~}& {\sqrt{(x + c)^2~+~y^2}} &{} \\ {}&{} & {~=~}& {\sqrt{(x + c)^2~+~b^2\left(1-\frac{x^2}{a^2} \right)}} &{} \\ {}&{} & {~=~}& {\sqrt{(x + c)^2~+~\left(a^2 - c^2 \right) \left(1-\frac{x^2}{a^2} \right)}~~ \color {green} {\text{- - - (I)}}} &{} \\ {}&{} & {~=~}& {\sqrt{x^2 + 2cx + c^2~+~a^2 - x^2 - c^2 + \frac{c^2 x^2}{a^2}}} &{} \\ {}&{} & {~=~}& {\sqrt{2cx~+~a^2 + \frac{c^2 x^2}{a^2}}} &{} \\ {}&{} & {~=~}& {\sqrt{\left(a+\frac{cx}{a} \right)^2}} &{} \\ {}&{} & {~=~}& {a+\frac{cx}{a}} &{} \\ \end{array}$

◼ Remarks:
• Line marked as (I):
In this line, we write a² - c² in the place of b².

5. Now we consider the distance of P from F₂. It can be written as:

$\begin{array}{ll} {}&{PF_2} & {~=~}& {\sqrt{(x~-~ c)^2~+~(y - 0)^2}} &{} \\ {}&{} & {~=~}& {\sqrt{(x - c)^2~+~y^2}} &{} \\ \end{array}$

6. As we did in the case of PF₁, here also, we substitute for y². We get:

$\begin{array}{ll} {}&{PF_2} & {~=~}& {\sqrt{(x - c)^2~+~y^2}} &{} \\ {}&{} & {~=~}& {\sqrt{(x - c)^2~+~b^2\left(1-\frac{x^2}{a^2} \right)}} &{} \\ {}&{} & {~=~}& {\sqrt{(x - c)^2~+~\left(a^2 - c^2 \right) \left(1-\frac{x^2}{a^2} \right)}~~ \color {green} {\text{- - - (I)}}} &{} \\ {}&{} & {~=~}& {\sqrt{x^2 - 2cx + c^2~+~a^2 - x^2 - c^2 + \frac{c^2 x^2}{a^2}}} &{} \\ {}&{} & {~=~}& {\sqrt{-2cx~+~a^2 + \frac{c^2 x^2}{a^2}}} &{} \\ {}&{} & {~=~}& {\sqrt{\left(a - \frac{cx}{a} \right)^2}} &{} \\ {}&{} & {~=~}& {a - \frac{cx}{a}} &{} \\ \end{array}$

◼ Remarks:
• Line marked as (I):
In this line, we write a² - c² in the place of b². 

7. So the sum of the distances of P from F₁ and F₂ is:
(PF₁ + PF₂) = (a + xc/a) + (a - xc/a) = 2a.

8. Consider step (5) below fig.11.34 at the beginning of this section. We saw that, sum of the distances of point B from the foci is '2a'.

9. So any point P(x,y) on the ellipse will satisfy the equation $\frac{x^2}{a^2}~+~\frac{y^2}{b^2}~=~1$
• The converse is proved.

◼ So we can write:
If the center of the ellipse is at O, major axis lies along the x-axis and minor axis lies along the y-axis, then equation of the ellipse is: $\frac{x^2}{a^2}~+~\frac{y^2}{b^2}~=~1$

---

Based on the above equation of the ellipse, we can write two interesting facts:
Fact 1:
This can be written in 5 steps:
1. We have: $\frac{x^2}{a^2}~+~\frac{y^2}{b^2}~=~1$
2. This can be rearranged as: $\frac{x^2}{a^2}~=~1~-~\frac{y^2}{b^2}$
• So $\frac{x^2}{a^2}$ will be always less than 1.
• That is: $\frac{x^2}{a^2}~\le~1$
⇒ $x^2 ~\le~a^2$
3. Solving the above inequality, we get:
    ♦ x should not be less than -a.
    ♦ x should not be greater than a.
• That is: $-a~\le~x~\le~a$
4. So we can write:
• Consider any point on the ellipse.
    ♦ The x-coordinate of that point will be greater than -a.
    ♦ The x-coordinate of that point will be less than a.
5. So the ellipse will lie between two vertical lines.
    ♦ The left vertical line is x = -a.
    ♦ The right vertical line is x = a.

Fact 2:
This can be written in 5 steps:
1. We have: $\frac{x^2}{a^2}~+~\frac{y^2}{b^2}~=~1$
2. This can be rearranged as: $\frac{y^2}{b^2}~=~1~-~\frac{x^2}{a^2}$
• So $\frac{y^2}{b^2}$ will be always less than 1.
• That is: $\frac{y^2}{b^2}~\le~1$
⇒ $y^2 ~\le~b^2$
3. Solving the above inequality, we get:
    ♦ y should not be less than -b.
    ♦ y should not be greater than b.
• That is: $-b~\le~y~\le~a$
4. So we can write:
• Consider any point on the ellipse.
    ♦ The y-coordinate of that point will be greater than -b.
    ♦ The y-coordinate of that point will be less than b.
5. So the ellipse will lie between two horizontal lines.
    ♦ The upper horizontal line is y = b.
    ♦ The lower horizontal line is y = -b.

• The ellipse and the lines in fig.11.35 below, demonstrates the two facts:

Fig.11.35

• Equation of the ellipse in the above fig. is: $\frac{x^2}{5^2}~+~\frac{y^2}{3^2}~=~1$
• We see that:
    ♦ Value of 'a' is 5.
        ✰ The vertical lines are related to 5.
    ♦ Value of 'b' is 3.
        ✰ The horizontal lines are related to 3.

---

• In this section, we saw a simplest equation of an ellipse.
    ♦ The major axis lies along the x-axis.
    ♦ The minor axis lies along the y-axis.
• We will get another simplest equation also when:
    ♦ The major axis lies along the y-axis.
    ♦ The minor axis lies along the x-axis.
• We will see it in the next section.

Previous

Contents

Next

Copyright©2023 Higher secondary mathematics.blogspot.com