EXERCISE-3

Eccentricity of an ellipse in which distance between their foci is $10$ and that of focus and corresponding directrix is $15$ is equal to:

An ellipse passes through the points $\left(-3,1\right)$ & $\left(2,-2\right)$ its principal axis are along the coordinate axes in order. The value of${\left(\mathrm{length} \mathrm{of} \mathrm{semi}-\mathrm{minor} \mathrm{axis}\right)}^2$  is equal to

If set of value(s) of $\mathrm{\alpha }$ for which the point $\left(7-\frac{5}{4}\alpha ,\alpha \right)$ lies inside the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ is $\left(\frac{12}{\lambda },\frac{16}{\lambda }\right)$, then value of $\lambda$ is

The tangent and normal to the ellipse $x^2+4y^2=4$ at a point $P\left(\theta \right)$ on it meet the major axis in $Q$ and $R$ respectively. If $QR=2$ and the eccentric angle $\theta$ of $P$ is given by $\cos \theta =\pm \left(\frac{a}{b}\right)$, where $a$ & $b$ are relative prime, then value of $\frac{b}{a}$ is equal to

If $3x+4y=12$ intersect the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ at $P$ and $Q$. If the point of intersection of tangents at $P$ and $Q$ is $\left(\frac{25}{a},\frac{16}{b}\right)$, then $a+b$ is equal to