The equations of the directrices of the ellipse $16{x^2} + 25{y^2} = 400$ are
$2x = \pm 25$
$5x = \pm 9$
$3x = \pm 10$
None of these
The tangent and normal to the ellipse $3x^2 + 5y^2 = 32$ at the point $P(2, 2)$ meet the $x-$ axis at $Q$ and $R,$ respectively. Then the area(in sq. units) of the triangle $PQR$ is
The line $x =8$ is the directrix of the ellipse $E: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ with the corresponding focus $(2,0)$. If the tangent to $E$ at the point $P$ in the first quadrant passes through the point $(0,4 \sqrt{3})$ and intersects the $x$-axis at $Q$, then $(3PQ)^2$ is equal to $........$
Let $T_1$ and $T_2$ be two distinct common tangents to the ellipse $E: \frac{x^2}{6}+\frac{y^2}{3}=1$ and the parabola $P: y^2=12 x$. Suppose that the tangent $T_1$ touches $P$ and $E$ at the point $A_1$ and $A_2$, respectively and the tangent $T_2$ touches $P$ and $E$ at the points $A_4$ and $A_3$, respectively. Then which of the following statements is(are) true?
($A$) The area of the quadrilateral $A_1 A _2 A _3 A _4$ is $35$ square units
($B$) The area of the quadrilateral $A_1 A_2 A_3 A_4$ is $36$ square units
($C$) The tangents $T_1$ and $T_2$ meet the $x$-axis at the point $(-3,0)$
($D$) The tangents $T_1$ and $T_2$ meet the $x$-axis at the point $(-6,0)$
Let $E$ be the ellipse $\frac{{{x^2}}}{9} + \frac{{{y^2}}}{4} = 1$ and $C$ be the circle ${x^2} + {y^2} = 9$. Let $P$ and $Q$ be the points $(1, 2)$ and $(2, 1)$ respectively. Then
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 :