Question

The length of the latus rectum of the ellipse $225x^{2}+125y^{2}=28125$ is}

• A. $\frac{50}{3}$
• B. $\frac{25}{3}$
• C. 5
• D. $\frac{50}{7}$
• E. $\frac{125}{3}$

Hint:

Math Tip: Always identify which axis is the major axis before applying the latus rectum formula. The formula is always $\frac{2 \times (\text{Minor Axis})^2}{\text{Major Axis}}$.

Answer 1

The Correct Option is A

Concept:
Coordinate Geometry - Properties of an Ellipse.
For an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, if $b>a$, the major axis is along the y-axis, and the length of the latus rectum is $\frac{2a^2}{b}$.
Step 1: Convert the given equation to standard form.
Given equation: $225x^2 + 125y^2 = 28125$
Divide the entire equation by $28125$ to make the right side equal to $1$: $$ \frac{225x^2}{28125} + \frac{125y^2}{28125} = 1 $$
Step 2: Simplify the fractions.
Divide the numerator and denominator: $$ \frac{x^2}{125} + \frac{y^2}{225} = 1 $$
Step 3: Identify the semi-major and semi-minor axes.
Comparing with $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$:

• $a^2 = 125$
• $b^2 = 225 \implies b = \sqrt{225} = 15$

Since $b^2>a^2$, this is a vertical ellipse (major axis along the y-axis).
Step 4: Calculate the length of the latus rectum.
Use the formula for the latus rectum of a vertical ellipse, $L = \frac{2a^2}{b}$: $$ L = \frac{2(125)}{15} $$ $$ L = \frac{250}{15} $$
Step 5: Simplify the fraction.
Divide the numerator and denominator by their greatest common divisor, which is 5: $$ L = \frac{50}{3} $$