An ellipse having foci at (3, 1) and (1, 1) p | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Let the foci be $S(3, 1)$ and $S'(1, 1)$,and the point on the ellipse be $P(1, 3)$.The distance $PS = \sqrt{(3-1)^2 + (1-3)^2} = \sqrt{2^2 + (-2)^

Vedclass · Accepted Answer

$\sqrt{2} - 1$

Vedclass · Answer

(A) Let the foci be $S(3, 1)$ and $S'(1, 1)$,and the point on the ellipse be $P(1, 3)$.The distance $PS = \sqrt{(3-1)^2 + (1-3)^2} = \sqrt{2^2 + (-2)^2} = \sqrt{4+4} = 2\sqrt{2}$.The distance $PS' = \sqrt{(1-1)^2 + (1-3)^2} = \sqrt{0^2 + (-2)^2} = 2$.By the definition of an ellipse,the sum of the distances from any point on the ellipse to the foci is constant and equal to the length of the major axis,$2a$.$2a = PS + PS' = 2\sqrt{2} + 2 \Rightarrow a = \sqrt{2} + 1$.The distance between the foci is $2ae = SS' = \sqrt{(3-1)^2 + (1-1)^2} = \sqrt{2^2 + 0^2} = 2$.Thus,$ae = 1$.Substituting $a = \sqrt{2} + 1$,we get $e = \frac{1}{\sqrt{2} + 1} = \frac{\sqrt{2} - 1}{(\sqrt{2} + 1)(\sqrt{2} - 1)} = \sqrt{2} - 1$.

An ellipse having foci at $(3, 1)$ and $(1, 1)$ passes through the point $(1, 3)$. Then its eccentricity is

Similar Questions

If the tangents on the ellipse $4x^2 + y^2 = 8$ at the points $(1, 2)$ and $(a, b)$ are perpendicular to each other,then $a^2$ is equal to

An ellipse having the coordinate axes as its axes and its major axis along the $Y$-axis,passes through the point $(-3, 1)$ and has eccentricity $e = \sqrt{\frac{2}{5}}$. Then its equation is:

If the latus rectum of an ellipse is half of its minor axis,then its eccentricity is ...

The equations of two ellipses are $\frac{x^2}{4}+\frac{y^2}{2}=1$ and $\frac{x^2}{36}+\frac{y^2}{b^2}=1$. If the product of their eccentricities is $\frac{\sqrt{2}}{3}$,then the product of the length of the major axis and minor axis of the second ellipse is $\qquad$

The angle between the tangents drawn from the point $(1, 2)$ to the ellipse $3x^2 + 2y^2 = 5$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module