Conic Sections

Let $S=\left\{(x,y)\in\mathbb N\times\mathbb N:9(x-3)^2+16(y-4)^2\le144\right\}$ and $T=\left\{(x,y)\in\mathbb R\times\mathbb R:(x-7)^2+(y-4)^2\le36\right\}$. Then $n(S\cap T)$ is equal to _______.

If foci of the hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}{12}=1$ and the ellipse $\dfrac{x^2}{25}+\dfrac{y^2}9=1$ coincide, then the eccentricity of the hyperbola, with foci $(0,\pm\sqrt{10})$ passing through the point (a, a + 1), is

Let the curves $y^2=kx$ and $xy=-1$ have a common tangent, whose slope is $\dfrac12$. Then k can NOT lie in the interval

The value of $\lambda^2$ is ____.

The value of $D$ is ____.

Let $E$ denote the parabola $y^2 = 8x$. Let $P = (-2, 4)$, and let $Q$ and $Q'$ be two distinct points on $E$ such that the lines $PQ$ and $PQ'$ are tangents to $E$. Let $F$ be the focus of $E$. Then which of the following statements is (are) TRUE?

The radius of the circle $C$ is ____.

The value of $\alpha$ is ____.

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 ____.

The locus of the mid-point of the line segment joining the focus of the parabola $y^2=4ax$ to a moving point of the parabola, is another parabola whose directrix 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

For a complex number $z$, let $\operatorname{Re}(z)$ denote the real part of $z$. Let $S$ be the set of all complex numbers $z$ satisfying $z^4 - |z|^4 = 4iz^2$, where $i = \sqrt{-1}$. Then the minimum possible value of $|z_1 - z_2|^2$, where $z_1, z_2 \in S$ with $\operatorname{Re}(z_1) > 0$ and $\operatorname{Re}(z_2) < 0$, is ______

Let $a$ and $b$ be positive real numbers such that $a > 1$ and $b < a$. Let $P$ be a point in the first quadrant that lies on the hyperbola $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$. Suppose the tangent to the hyperbola at $P$ passes through the point $(1, 0)$, and suppose the normal to the hyperbola at $P$ cuts off equal intercepts on the coordinate axes. Let $\Delta$ denote the area of the triangle formed by the tangent at $P$, the normal at $P$ and the $x$-axis. If $e$ denotes the eccentricity of the hyperbola, then which of the following statements is/are TRUE?

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?

Which of the following is the only INCORRECT combination?

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?

Let $H: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, where $a > b > 0$, be a hyperbola in the $xy$-plane whose conjugate axis $LM$ subtends an angle of $60^\circ$ at one of its vertices $N$. Let the area of the triangle $LMN$ be $4\sqrt{3}$.

The correct option is:

If $2x - y + 1 = 0$ is a tangent to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{16} = 1$, then which of the following CANNOT be sides of a right angled triangle?

If a chord, which is not a tangent, of the parabola $y^2 = 16x$ has the equation $2x + y = p$, and midpoint $(h, k)$, then which of the following is(are) possible value(s) of $p$, $h$ and $k$?

For $a = \sqrt{2}$, if a tangent is drawn to a suitable conic (Column 1) at the point of contact $(-1, 1)$, then which of the following options is the only CORRECT combination for obtaining its equation?