Question

The length of the latus rectum of the ellipse \[ \frac{x^2}{9}+\frac{y^2}{16}=1 \] is ________.

• A. $\frac{3}{2}$
• B. $8$
• C. $\frac{9}{2}$
• D. $2$
• E. $\frac{25}{2}$

Hint:

The Latus Rectum formula is $2 \times \frac{\text{(minor semi-axis)}^2}{\text{major semi-axis}}$.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
We need to find the length of the latus rectum of an ellipse given its standard equation. The first step is to identify the semi-major axis (a) and the semi-minor axis (b) from the equation.
Step 2: Key Formula or Approach
The standard equation of an ellipse centered at the origin is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) or \(\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1\), where \(a\) is the semi-major axis and \(b\) is the semi-minor axis (\(a>b\)).
The length of the latus rectum (L.R.) is given by the formula:
\[ L.R. = \frac{2b^2}{a} \] Step 3: Detailed Explanation
1. Analyze the equation of the ellipse.
The given equation is \( \frac{x^2}{9} + \frac{y^2}{16} = 1 \).
We compare this with the standard forms. We see that the denominator of the \(y^2\) term (16) is greater than the denominator of the \(x^2\) term (9).
This means the major axis of the ellipse is along the y-axis.
So, the equation is of the form \(\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1\).
2. Identify \(a^2\) and \(b^2\).
From the comparison:
- \(a^2 = 16 \implies a = \sqrt{16} = 4\) (semi-major axis)
- \(b^2 = 9 \implies b = \sqrt{9} = 3\) (semi-minor axis)
3. Calculate the length of the latus rectum.
Using the formula \(L.R. = \frac{2b^2}{a}\):
\[ L.R. = \frac{2 \times 9}{4} \] \[ L.R. = \frac{18}{4} \] Simplify the fraction:
\[ L.R. = \frac{9}{2} \] Step 4: Final Answer
The length of the latus rectum is \(\frac{9}{2}\).