Question

If the area of the Ellipse is \(\frac{x^2}{25}+\frac{y^2}{\lambda^2}=1\) is 20π square units, then λ is

• A. ±4
• B. ±3
• C. ±2
• D. ±1

Answer 1

The Correct Option is A

The equation of the ellipse is given by: \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \] where \( a \) and \( b \) are the lengths of the semi-major and semi-minor axes, respectively. The area of the ellipse is given by the formula: \[ \text{Area} = \pi \cdot a \cdot b \] From the equation, we can identify \( a^2 = 25 \), which means \( a = 5 \). Given that the area of the ellipse is \( 20 \pi \) square units, we can substitute into the area formula: \[ \pi \cdot 5 \cdot \lambda = 20 \pi \] Dividing both sides by \( \pi \), we get: \[ 5 \cdot \lambda = 20 \] Solving for \( \lambda \), we find: \[ \lambda = 4 \] 

The correct answer is (A) : ±4.

Answer 2

Given the equation of the ellipse:

\[ \frac{x^2}{25} + \frac{y^2}{\lambda^2} = 1 \]

Step 1: Identify the semi-major and semi-minor axes

Here, the standard form of an ellipse is: \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \] where the area is given by: \[ \text{Area} = \pi a b \] In our case, \( a^2 = 25 \Rightarrow a = 5 \), and \( b^2 = \lambda^2 \Rightarrow b = |\lambda| \)

Step 2: Use the given area

\[ \pi \cdot 5 \cdot |\lambda| = 20\pi \Rightarrow 5 |\lambda| = 20 \Rightarrow |\lambda| = 4 \Rightarrow \lambda = \pm 4\]

Final Answer: \( \boxed{\pm 4} \)