Ellipse

Let $E$ be the ellipse $\dfrac{x^2}{16} + \dfrac{y^2}{9} = 1$. For any three distinct points $P$, $Q$ and $Q'$ on $E$, let $M(P, Q)$ be the mid-point of the line segment joining $P$ and $Q$, and $M(P, Q')$ be the mid-point of the line segment joining $P$ and $Q'$. Then the maximum possible value of the distance between $M(P, Q)$ and $M(P, Q')$, as $P$, $Q$ and $Q'$ vary on $E$, is ____.

Let $a, b$ and $\lambda$ be positive real numbers. Suppose $P$ is an end point of the latus rectum of the parabola $y^2 = 4\lambda x$, and suppose the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ passes through the point $P$. If the tangents to the parabola and the ellipse at the point $P$ are perpendicular to each other, then the eccentricity of the ellipse is

Define the collections $\{E_1, E_2, E_3, \ldots\}$ of ellipses and $\{R_1, R_2, R_3, \ldots\}$ of rectangles as follows:

$E_1: \dfrac{x^2}9 + \dfrac{y^2}4 = 1$;

$R_1$: rectangle of largest area, with sides parallel to the axes, inscribed in $E_1$;

$E_n$: ellipse $\dfrac{x^2}{a_n^2} + \dfrac{y^2}{b_n^2} = 1$ of largest area inscribed in $R_{n-1}$, $n > 1$;

$R_n$: rectangle of largest area, with sides parallel to the axes, inscribed in $E_n$, $n > 1$.

Then which of the following options is/are correct?

Consider two straight lines, each of which is tangent to both the circle $x^2 + y^2 = \frac{1}{2}$ and the parabola $y^2 = 4x$. Let these lines intersect at the point $Q$. Consider the ellipse whose center is at the origin $O(0, 0)$ and whose semi-major axis is $OQ$. If the length of the minor axis of this ellipse is $\sqrt{2}$, then which of the following statement(s) is (are) TRUE?

The tangent to a suitable conic (Column 1) at $\left(\sqrt{3}, \frac{1}{2}\right)$ is found to be $\sqrt{3}x + 2y = 4$, then which of the following options is the only CORRECT combination?

The eccentricity of an ellipse whose centre is at the origin is $\dfrac12$. If one of its directrices is $x=-4$, then the equation of the normal to it at $\left(1,\dfrac32\right)$ is :

Consider an ellipse, whose centre is at the origin and its major axis is along the $x$-axis. If its eccentricity is $\dfrac35$ and the distance between its foci is 6, then the area (in sq. units) of the quadrilateral inscribed in the ellipse, with the vertices as the vertices of the ellipse, is :

The eccentricity of an ellipse having centre at the origin, axes along the co-ordinate axes and passing through the points (4, −1) and (−2, 2) is :

The orthocentre of the triangle $F_1MN$ is

If the tangents to the ellipse at $M$ and $N$ meet at $R$ and the normal to the parabola at $M$ meets the $x$-axis at $Q$, then the ratio of area of the triangle $MQR$ to area of the quadrilateral $MF_1NF_2$ is

If the tangent at a point on the ellipse $\dfrac{x^2}{27}+\dfrac{y^2}3=1$ meets the coordinate axes at A and B, and O is the origin, then the minimum area (in sq. units) of the triangle OAB is :

A hyperbola whose transverse axis is along the major axis of the conic, $\dfrac{x^2}3+\dfrac{y^2}4=4$ and has vertices at the foci of this conic. If the eccentricity of the hyperbola is $\dfrac32$, then which of the following points does NOT lie on it ?

Suppose that the foci of the ellipse $\dfrac{x^2}9+\dfrac{y^2}5=1$ are $(f_1,0)$ and $(f_2,0)$ where $f_1>0$ and $f_2<0$. Let $P_1$ and $P_2$ be two parabolas with a common vertex at $(0,0)$ and with foci at $(f_1,0)$ and $(2f_2,0)$, respectively. Let $T_1$ be a tangent to $P_1$ which passes through $(2f_2,0)$ and $T_2$ be a tangent to $P_2$ which passes through $(f_1,0)$. If $m_1$ is the slope of $T_1$ and $m_2$ is the slope of $T_2$, then the value of $\left(\dfrac1{m_1^2}+m_2^2\right)$ is

Let $E_1$ and $E_2$ be two ellipses whose centers are at the origin. The major axes of $E_1$ and $E_2$ lie along the $x$-axis and the $y$-axis, respectively. Let $S$ be the circle $x^2+(y-1)^2=2$. The straight line $x+y=3$ touches the curves $S$, $E_1$ and $E_2$ at $P$, $Q$ and $R$, respectively. Suppose that $PQ=PR=\dfrac{2\sqrt2}3$. If $e_1$ and $e_2$ are the eccentricities of $E_1$ and $E_2$, respectively, then the correct expression(s) is(are)

The area (in sq. units) of the quadrilateral formed by the tangents at the end points of the latera recta to the ellipse $\dfrac{x^2}9+\dfrac{y^2}5=1$, is :

An ellipse passes through the foci of the hyperbola, $9x^2-4y^2=36$ and its major and minor axes lie along the transverse and conjugate axes of the hyperbola respectively. If the product of eccentricities of the two conics is $\dfrac12$, then which of the following points does not lie on the ellipse ?

If the distance between the foci of an ellipse is half the length of its latus rectum, then the eccentricity of the ellipse is :

The locus of the foot of perpendicular drawn from the centre of the ellipse $x^2+3y^2=6$ on any tangent to it is :

If OB is the semi-minor axis of an ellipse, $F_1$ and $F_2$ are its foci and the angle between $F_1B$ and $F_2B$ is a right angle, then the square of the eccentricity of the ellipse is :

A stair-case of length $l$ rests against a vertical wall and a floor of a room,. Let P be a point on the stair-case, nearer to its end on the wall, that divides its length in the ratio 1 : 2. If the stair-case begins to slide on the floor, then the locus of P is :