If the chords of contact of tangents drawn from two points $(x_1, y_1)$ and $(x_2, y_2)$ to the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ are at right angles,then $\frac{x_1 x_2}{y_1 y_2} = \dots$

If the length of the major axis of an ellipse is three times the length of its minor axis,then its eccentricity is

Let $E_1 = \frac{x^2}{9} + \frac{y^2}{4} = 1$ and $E_2 = \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ be two ellipses and $R$ be a rectangle with sides parallel to the coordinate axes. Let $E_1$ be the inscribed ellipse in $R$ and $E_2$ be the circumscribed ellipse on $R$. If $E_2$ passes through $(0, 4)$,then:

If tangents are drawn to the ellipse $x^2+2y^2=2$,then the locus of the midpoints of the intercepts made by the tangents between the coordinate axes is

An arch is in the form of a semi-ellipse. It is $8 \, m$ wide and $2 \, m$ high at the centre. Find the height of the arch at a point $1.5 \, m$ from one end. (in $, m$)

$A$ particle is travelling in a clockwise direction on the ellipse $\frac{x^2}{100} + \frac{y^2}{25} = 1$. If the particle leaves the ellipse at the point $(-8, 3)$ and travels along the tangent to the ellipse at that point,then the point where the particle crosses the $Y$-axis is: