Question

For the ellipse $25{x^2} + 9{y^2} - 150x - 90y + 225 = 0$ , the eccentricity is equal to :
1. $\dfrac{2}{5}$
2. $\dfrac{3}{5}$
3. $\dfrac{{\sqrt {15} }}{{24}}$
4. $\dfrac{4}{5}$

Answer 1

Hint : This question requires one to use the formula of eccentricity of an ellipse in the standard form i.e., for the ellipse $\dfrac{{{{(x - \alpha )}^2}}}{{{a^2}}} + \dfrac{{{{(y - \beta )}^2}}}{{{b^2}}} = 1$ , the eccentricity is given as
${b^2} = {a^2}(1 - {e^2})$ , where b is the length of the semi major axis.

Complete step-by-step answer :
Let’s first convert $25{x^2} + 9{y^2} - 150x - 90y + 225 = 0$ into a standard format,
$\Rightarrow 25{x^2} + 9{y^2} - 150x - 90y + 225 = 0$
Now, taking 25 common from ${x^2}$ and 9 common from ${y^2}$ we get
$\Rightarrow 25({x^2} - 6x + 9) + 9({y^2} - 10y) = 0$
Now, converting the ${y^2}$ into a perfect square we need to add $900$ to both the sides of the equation
$\Rightarrow 25({x^2} - 6x + 9) + 9({y^2} - 10y + 100) = 900$
Condensing the algebraic expressions into whole squares using algebraic identities, we get,
$\Rightarrow 25{(x - 3)^2} + 9{(y - 10)^2} = 900$
Now, dividing both the sides by $900$ we get
$\Rightarrow \dfrac{{{{(x - 3)}^2}}}{{36}} + \dfrac{{{{(y - 10)}^2}}}{{100}} = 1$
Now, let’s apply the eccentricity formula
$\Rightarrow {b^2} = {a^2}(1 - {e^2})$
Substituting the values as ${a^2} = 100$ and ${b^2} = 36$
$\Rightarrow 36 = 100(1 - {e^2})$
Now, shifting the terms in the equation and cancelling the common factors, we get,
$\Rightarrow \dfrac{9}{{25}} = 1 - {e^2}$
Simplifying the expressions, we get,
$\Rightarrow {e^2} = 1 - \dfrac{9}{{25}}$
$\Rightarrow {e^2} = \dfrac{{16}}{{25}}$
Taking square root on both sides of the equation, we get,
$\Rightarrow {e^{}} = \dfrac{4}{5}$
Thus, the eccentricity of the given ellipse is $\dfrac{4}{5}$ .
Therefore, option(4) is the correct answer.
So, the correct answer is “Option 4”.

Note : This question can be very exhausting if calculations are not done properly. Take in account the signs of the terms correctly. The formula should not be applied in the wrong sense. care should be taken while carrying out the calculations and using algebraic identities.