Question

The latus rectum of an ellipse is 10 and the minor axis is equal to the distance between the foci. Equation 1 of the ellipse is-
A) ${x^2} + 2{y^2} = 100$
B) ${x^2} + \sqrt 2 {y^2} = 10$
C) ${x^2} - 2{y^2} = 100$
D) None of these

Answer 1

We will first consider the equation of ellipse $\dfrac{{{x^2}}}{{{a^2}}} + \dfrac{{{y^2}}}{{{b^2}}} = 1$. We will first put the value of latus rectum equal to $\dfrac{{2{b^2}}}{a}$ and determine the value of ${b^2}$ from it. Next, we are given that minor axis is equal to the distance between the foci so, we will first find the distance between the foci and then put the minor axis equal to the distance obtained and evaluate the value of ${a^2}$ from it. Using the value of ${a^2}$ we can find the value of ${b^2}$ and substitute in the general form of the equation of the ellipse.

Complete step by step solution: We will consider the given data that is the latus rectum of an ellipse is equal to 10 and the minor axis is equal to the distance between the foci.
We need to find the equation of the ellipse.
We will first consider the general form of the equation of the ellipse.
That is $\dfrac{{{x^2}}}{{{a^2}}} + \dfrac{{{y^2}}}{{{b^2}}} = 1$
Now, the objective is to determine the value of ${a^2}$ and ${b^2}$ and substitute it in the general form.
Since we are given that length of latus rectum is 10 And formula of the length of latus rectum is given by $\dfrac{{2{b^2}}}{a}$.
Thus, we get,

$$\Rightarrow \dfrac{{2{b^2}}}{a} = 10 \\\ \Rightarrow {b^2} = 5a \\\$$

Now, as we know that the coordinates of foci are $\left( {ae,0} \right)$ and $\left( { - ae,0} \right)$.
We can find the distance between the foci by using the distance formula, $d = \sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}}$
Thus, we get,

$$\Rightarrow d = \sqrt {{{\left( { - ae - ae} \right)}^2} + {{\left( {0 - 0} \right)}^2}} \\\ \Rightarrow d = \sqrt {{{\left( { - 2ae} \right)}^2}} \\\ \Rightarrow d = \sqrt {4{a^2}{e^2}} \\\ \Rightarrow d = 2ae \\\$$

As we are given in the question that length of the minor axis that is $2b$ and the distance between the foci that is $2ae$ are equal.
Thus, we get,

$$\Rightarrow 2b = 2ae \\\ \Rightarrow b = ae \\\$$

Since, we know that, ${b^2} = {a^2}\left( {1 - {e^2}} \right)$, then we will substitute the value of $b = ae$ in it.
Therefore,

$$\Rightarrow {a^2}{e^2} = {a^2}\left( {1 - {e^2}} \right) \\\ \Rightarrow {a^2}{e^2} = {a^2} - {a^2}{e^2} \\\ \Rightarrow 2{a^2}{e^2} = {a^2} \\\ \Rightarrow e = \dfrac{1}{{\sqrt 2 }} \\\$$

Now, by substituting the value of $e$ we can calculate the value of $b$.
Thus, we get,
$b = \dfrac{a}{{\sqrt 2 }}$
Next, we will Square both sides and substitute the value of ${b^2}$ in the expression.
Thus, we have,

$$\Rightarrow {b^2} = \dfrac{{{a^2}}}{2} \\\ \Rightarrow 5a = \dfrac{{{a^2}}}{2} \\\ \Rightarrow a = 10 \\\ \Rightarrow {a^2} = 100 \\\$$

Thus, we can find the value of ${b^2}$ also.

$${b^2} = 5\left( {10} \right) \\\ = 50 \\\$$

Hence, we will substitute the value of ${a^2}$ and ${b^2}$ in the general form.
Thus, we get,

$$\Rightarrow \dfrac{{{x^2}}}{{100}} + \dfrac{{{y^2}}}{{50}} = 1 \\\ \Rightarrow {x^2} + 2{y^2} = 100 \\\$$

Hence, the equation 1 of the ellipse is ${x^2} + 2{y^2} = 100$.

Thus, option A is correct.

Note: Note: We need to remember the distance formula to find the distance between both the foci. We have used the general values of the equation of the ellipse and find the values of ${a^2}$ and ${b^2}$. In equation 1 of the ellipse, points of foci lie on the $x$-axis and they are at equidistant from each other. Remember to find the foci points as the length of the minor axis is equal to the distance between both the foci.