If OT is the semi-minor axis of an ellipse,A | vedclass

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

Question

(C) Let the equation of the ellipse be $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. The coordinates of the foci are $A(-ae, 0)$ and $B(ae, 0)$,and the end

Vedclass · Accepted Answer

$\frac{1}{\sqrt{2}}$

Vedclass · Answer

(C) Let the equation of the ellipse be $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. The coordinates of the foci are $A(-ae, 0)$ and $B(ae, 0)$,and the endpoint of the semi-minor axis is $T(0, b)$. The slope of $AT$ is $m_1 = \frac{b - 0}{0 - (-ae)} = \frac{b}{ae}$. The slope of $BT$ is $m_2 = \frac{b - 0}{0 - ae} = -\frac{b}{ae}$. Since $\angle ATB = 90^{\circ}$,the product of the slopes is $-1$: $\left(\frac{b}{ae}\right) \left(-\frac{b}{ae}\right) = -1$. $\Rightarrow \frac{b^2}{a^2 e^2} = 1$ $\Rightarrow b^2 = a^2 e^2$. We know that for an ellipse,$b^2 = a^2(1 - e^2)$. Substituting $b^2 = a^2 e^2$ into the equation: $a^2 e^2 = a^2(1 - e^2)$. $e^2 = 1 - e^2 $ $\Rightarrow 2e^2 = 1 $ $\Rightarrow e^2 = \frac{1}{2} $ $\Rightarrow e = \frac{1}{\sqrt{2}}$.

If $OT$ is the semi-minor axis of an ellipse,$A$ and $B$ are its foci and $\angle ATB$ is a right angle,then the eccentricity of that ellipse is

Similar Questions

Find the coordinates of the foci,the vertices,the length of the major axis,the minor axis,the eccentricity,and the length of the latus rectum of the ellipse $16x^{2} + y^{2} = 16$.

The eccentricity of the curve represented by the equation ${x^2} + 2{y^2} - 2x + 3y + 2 = 0$ is

The equation $\frac{x^2}{2-r}+\frac{y^2}{r-5}+1=0$ represents an ellipse if

The length of the chord of the ellipse $\frac{x^2}{4} + y^2 = 1$ formed on the line $y = x + 1$ is

$A$ staircase of length $l$ rests against a vertical wall and a floor of a room. Let $P$ be a point on the staircase,nearer to its end on the wall,that divides its length in the ratio $1 : 2$. If the staircase begins to slide on the floor,then the locus of $P$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module