The distance of a focus of the ellipse $9x^2+16y^2 =144$ from an end of the minor axis is
- $\frac{3}{2}$
- 3
- 4
- None of these
The Correct Option is C
Solution and Explanation
$ a^{2}= 16, b^{2}=9$. Focus is $\left(ae, 0\right)$ i.e., $\left(4e, 0\right) $
$b^{2} = a^{2}\left(1-e^{2}\right) $ gives $\frac{9}{16} = 1-e^{2} $
$\Rightarrow e^{2}=1 -\frac{9}{16} = \frac{7}{16} $
$ \Rightarrow e= \frac{\sqrt{7}}{4} $
One end of mirror axis is $B\left(0, 3\right)$
Distance of the focus from this end
$ \sqrt{\left(4e-0\right)^{2}+\left(0-3\right)^{2}} = \sqrt{16e^{2}+9}$
$\sqrt{\frac{16\times7}{16}+9} = \sqrt{7+9} $
$ = \sqrt{16} = 4$
Top Questions on Ellipse
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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}