Find the equation of the ellipse which passes through the points $(-3, 1)$ and $(2, -2)$,whose center lies at $(0, 0)$ and major axis lies along the $X$-axis.

Find the equation for the ellipse that satisfies the given conditions: Vertices $(\pm 6, 0)$,foci $(\pm 4, 0)$.

The line $lx + my + n = 0$ will cut the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ at points whose eccentric angles differ by $\pi/2$ if:

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:

Statement $-1$: If two tangents are drawn to an ellipse from a single point and if they are perpendicular to each other,then the locus of that point is always a circle.
Statement $-2$: For an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$,the locus of the point from which two perpendicular tangents are drawn is $x^2 + y^2 = a^2 + b^2$.

The equations of the tangents drawn at the ends of the major axis of the ellipse $9x^2 + 5y^2 - 30y = 0$ are: