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(D) The equation of the ellipse is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$.The coordinates of the ends of the latus rectum are $(\pm ae, \pm \frac{b^2

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both $(A)$ and $(B)$

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(D) The equation of the ellipse is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$.The coordinates of the ends of the latus rectum are $(\pm ae, \pm \frac{b^2}{a})$.The equation of the tangent at $(ae, \frac{b^2}{a})$ is $\frac{x(ae)}{a^2} + \frac{y(b^2/a)}{b^2} = 1$.Simplifying this,we get $\frac{ex}{a} + \frac{y}{a} = 1$,which is $ex + y = a$.Since $e^2 = 1 - \frac{b^2}{a^2}$,we have $e = \sqrt{1 - \frac{b^2}{a^2}}$.Substituting $e$ into the equation: $\sqrt{1 - \frac{b^2}{a^2}} x + y = a$.For the tangent to pass through a fixed point regardless of $b$,we observe the family of lines. The tangents at the four ends of the latus rectum are given by $\pm ex \pm y = a$.These lines pass through the fixed points $(0, a)$ and $(0, -a)$.

If a number of ellipses are described having the same major axis $2a$ but a variable minor axis,then the tangents at the ends of their latera recta pass through fixed points which are:

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