If the area of the Ellipse is $\frac{x^{2}}{25}+\frac{y^{2}}{\lambda^{2}}=1$ is $20 \pi$ square units, then $\lambda$ is
- $\pm 4$
- $\pm 3$
- $\pm 2$
- $\pm 1$
The Correct Option is A
Solution and Explanation
To find the value of $\lambda$, we need to first understand the equation of the ellipse given and its relation to the area.
The general form of the ellipse equation is:
$\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$
Here, $a^2 = 25$ (as given) and $b^2 = \lambda^2$.
The area of an ellipse is calculated using the formula:
$\text{Area} = \pi \times a \times b$
Given that the area of the ellipse is $20\pi$ square units, we can set up the equation:
$\pi \times \sqrt{25} \times \lambda = 20\pi$
Simplifying this, we have:
$5\lambda = 20$
Solving for $\lambda$:
$\lambda = \frac{20}{5} = 4$
Therefore, $\lambda$ is $4$. Since the problem asks for both positive and negative values, the answer is $\pm 4$.
Thus, the correct answer is:
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Concepts Used:
Ellipse
Ellipse Shape
An ellipse is a locus of a point that moves in such a way that its distance from a fixed point (focus) to its perpendicular distance from a fixed straight line (directrix) is constant. i.e. eccentricity(e) which is less than unity
Properties
- Ellipse has two focal points, also called foci.
- The fixed distance is called a directrix.
- The eccentricity of the ellipse lies between 0 to 1. 0≤e<1
- The total sum of each distance from the locus of an ellipse to the two focal points is constant
- Ellipse has one major axis and one minor axis and a center
Read More: Conic Section
Eccentricity of the Ellipse
The ratio of distances from the center of the ellipse from either focus to the semi-major axis of the ellipse is defined as the eccentricity of the ellipse.
The eccentricity of ellipse, e = c/a
Where c is the focal length and a is length of the semi-major axis.
Since c ≤ a the eccentricity is always greater than 1 in the case of an ellipse.
Also,
c2 = a2 – b2
Therefore, eccentricity becomes:
e = √(a2 – b2)/a
e = √[(a2 – b2)/a2] e = √[1-(b2/a2)]
Area of an ellipse
The area of an ellipse = πab, where a is the semi major axis and b is the semi minor axis.
Position of point related to Ellipse
Let the point p(x1, y1) and ellipse
(x2 / a2) + (y2 / b2) = 1
If [(x12 / a2)+ (y12 / b2) − 1)]
= 0 {on the curve}
<0{inside the curve}
>0 {outside the curve}