Let the eccentricity of an ellipse \(\frac {x^2}{a^2}+\frac {y^2}{b^2}=1\), \(a>b\), be \(\frac 14\). If this ellipse passes through the point \((−4\sqrt {\frac 25},3)\), then \(a^2 + b^2\) is equal to :
Let the eccentricity of an ellipse \(\frac {x^2}{a^2}+\frac {y^2}{b^2}=1\), \(a>b\), be \(\frac 14\). If this ellipse passes through the point \((−4\sqrt {\frac 25},3)\), then \(a^2 + b^2\) is equal to :
- 29
- 31
- 32
- 34
The Correct Option is B
Solution and Explanation
\(\frac {x^2}{a^2}+\frac {y^2}{b^2}=1\)
\(⇒ \frac {(−4\sqrt {\frac 25})^2}{a^2}+\frac {3^2}{b^2}=1\)
\(⇒ \frac {32}{a^2}+\frac {9}{b^2}=1\) ....(i)
\(a^2(1-e^2)=b^2\)
\(⇒ a^2(1-\frac {1}{16}) = b^2\)
\(⇒ 15a^2 = 16 b^2\)\( ⇒ a^2 = \frac {16b^2}{15}\)
From (i)
\(\frac {6}{b^2}+\frac {9}{b^2}=1\)
\(⇒ b^2 = 15\)
and \(a^2 = 16\)
\(⇒ a^2+b^2 = 15+16\) \(= 31\)
So, the correct option is (B): \(31\)
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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}