The latus rectum of an ellipse is $10$ and the minor axis is equal to the distance between the foci. The equation of the ellipse is
${x^2} + 2{y^2} = 100$
${x^2} + \sqrt 2 {y^2} = 10$
${x^2} - 2{y^2} = 100$
None of these
The normal at a variable point $P$ on an ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}}= 1$ of eccentricity e meets the axes of the ellipse in $ Q$ and $R$ then the locus of the mid-point of $QR$ is a conic with an eccentricity $e' $ such that :
A triangle is formed by the tangents at the point $(2,2)$ on the curves $y^2=2 x$ and $x^2+y^2=4 x$, and the line $x+y+2=0$. If $r$ is the radius of its circumcircle, then $r ^2$ is equal to $........$.
An ellipse is inscribed in a circle and a point is inside a circle is choosen at random. If the probability that this point lies outside the ellipse is $\frac {2}{3}$ then eccentricity of ellipse is $\frac{{a\sqrt b }}{c}$ . Where $gcd( a, c) = 1$ and $b$ is square free integer ($b$ is not divisible by square of any integer except $1$ ) then $a · b · c$ is
If a number of ellipse be 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 can be
Equation of the ellipse with eccentricity $\frac{1}{2}$ and foci at $( \pm 1,\;0)$ is