Conic Sections - Class 11

Let $M$ be the $3 \times 3$ symmetric matrix associated with the conic $S \equiv 2xy + 4x + 6y = 0$, where the conic is represented as $\mathbf{x}^T M \mathbf{x} = 0$ with $\mathbf{x} = \begin{pmatrix} x \\ y \\ 1 \end{pmatrix}$. Let $C$ be the cofactor matrix of $M$. A line $L$ is defined by the equation $\begin{pmatrix} x & y & 1 \end{pmatrix} C \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} = 0$. If $(x_0, y_0)$ is the pole of the line $L$ with respect to the conic $S=0$, then find the coordinates $(x_0, y_0)$.

Let $z_1, z_2, z_3, z_4$ be the distinct roots of the equation $z^4 - 1 = 0$. Let $z_a$ and $z_b$ be the roots satisfying $\text{Re}(z) = 0$. Let $C$ be the locus of a point $z$ in the Argand plane satisfying $|z - z_a| + |z - z_b| = 4$. A real-valued function $f: D \to \mathbb{R}$ is defined as $f(x) = \max \{ \text{Im}(z) : \text{Re}(z) = x, z \in C \}$, where $D$ is the maximal domain for which $f$ is defined. Let $D_1 \subset D$ be the largest interval $[p, q]$ such that the restriction of $f$ to $D_1$ is an injective function and $f'(x) > 0$ for $x \in (p, q)$. Find the value of $\int_{D_1} (f(x))^2 dx$.

Let $C_e$ be the conic $x^2 + 2exy + y^2 = e^2$, where $e \in (0,1)$ is a parameter. Let $A(e)$ be the area of the region in the first quadrant bounded by $C_e$ and the coordinate axes. The value of $e$ that maximizes the function $F(e) = \frac{\sqrt{1-e^2}}{e^2} A(e) + \frac{1}{2} \ln(e)$ is:

Let $C$ be an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$) with eccentricity $e$. The value of $e^2$ is given by the limit: $e^2 = \lim_{u \to 0} \frac{u^3 \sin\left(\frac{1}{u}\right) + \ln\left(1 + \frac{3}{4}u^2\right)}{u^2}$ Let $f(t) = (t^2 - 5t + 5)^{\ln(t^2 - 5t + 5)}$ and $S$ be the set of all critical points of $f(t)$ in its domain. Let $P(x_0, y_0)$ be a point on $C$ in the first quadrant. The normal to $C$ at $P$ intersects the x-axis at $G(X_N, 0)$, where $X_N$ is the sum of all elements in $S$. If the tangent to $C$ at $P$ intersects the x-axis at $T(X_T, 0)$ with $X_T = 12$, find the value of $9y_0^2$.

Let $P$ be the parabola $x^2 - 2x + 4y - 11 = 0$ and $AB$ be its chord of contact drawn from the point $M(4, 1)$. Let $S$ be the region bounded by the parabola $P$ and the chord $AB$. Evaluate the integral $E = 540 \iint_S \left[ \frac{x}{2} \right] dy \, dx$ where $[ \cdot ]$ denotes the greatest integer function.

Let $\Delta_1(t) = \begin{vmatrix} 1 & t & t^2 \\ 1 & 1 & 1 \\ 1 & 2 & 4 \end{vmatrix}$ and $\Delta_2(t) = \begin{vmatrix} 1 & t & t^2 \\ 1 & 3 & 9 \\ 1 & 4 & 16 \end{vmatrix}$. Let $S$ be the set of all integers $t \in [0, 10]$ that satisfy the inequality $\frac{\Delta_1(t)}{\Delta_2(t)} \ge 1$. Let $h = n(S)$ (the number of elements in $S$) and $k = \sum_{t \in S} t$. A parabola $P$ has its vertex at $(h, k)$ and passes through the point $(h+2, k+12)$. The $y$-intercept of the parabola $P$ is:

Let $S$ be the set of all continuous functions $f: [1, 3] \to \mathbb{R}$ satisfying $f(x) \ge \left| \int_{x}^{x^2-4x+5} (t-2) \, dt \right|$ for all $x \in [1, 3]$. The minimum possible value of the definite integral $\int_{1}^{3} f(x) \, dx$ is:

Let $S$ be the set of all real numbers $t$ such that the vertex of the parabola $P_t: y = x^2 - 2tx + t^6 - 21t^4 + 85t^2 - 64$ lies on the $x$-axis, and the line $L: y = -2\\sqrt{39}x - 39$ does not intersect $P_t$. Find the value of $\\sum_{t \\in S} t$.

Let $E$ be the set of all values attained by the function $f(x) = \frac{x^2}{2} - 4|x|$ at its points of local extrema, and let $k = n(E)$ be the number of elements in $E$. For each $m \in E$, let $L_m$ be the line with slope $m$ which is a valid tangent to the parabola $y^2 = 4x$. If $L_m$ intersects the latus rectum of the parabola at $Q_m(1, y_m)$, find the sum of the values of $(m^2 y_m^2 + k)$ over all valid lines $L_m$.

Let $L$ be the latus rectum of the parabola $y^2 = 4x$. Consider the integral evaluated along $L$: $I = \int_{-2}^{2} \frac{y}{y^2+4} \ln\left( \frac{y^4 - 24y^2 + 16}{(y^2+4)^2} \right) dy$ Which of the following represents the correct evaluation of $I$?

Let $E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$) be an ellipse with eccentricity $e$. Let $P(ae, y_1)$ be a point on $E$ in the first quadrant. A matrix $M$ is defined as $M = \begin{pmatrix} a & 2y_1 \\ 2y_1 & a e^2 \end{pmatrix}$ Let $S$ be the set of all possible values of $e$ for which $M$ is a singular matrix. The sum of the elements of $S$ is:

Let $E$ be the ellipse $\frac{x^2}{36} + \frac{y^2}{12} = 1$ in the $xy$-plane, and let $C$ be its auxiliary circle. Let $L$ be the chord of $C$ joining the points having eccentric angles $0$ and $\frac{\pi}{2}$. If $P$ is the pole of the line $L$ with respect to the ellipse $E$, and $P'$ is the image of $P$ in the plane $\Pi: x + y + z = 14$, find the square of the distance from $P'$ to the center of $C$.

Let $E$ be an ellipse whose auxiliary circle has an area of $16\pi$ and the distance between its directrices is $16$. Let $\alpha$ and $\beta$ be the roots of the quadratic equation $x^2 - \frac{a}{2}x + a^2 e^2 = 0$, where $a$ is the semi-major axis and $e$ is the eccentricity of $E$. A sequence is defined as $T_k = \alpha^k + \beta^k$ for integers $k \ge 0$. Find the value of $\frac{1}{2^{11}} \sum_{k=0}^{10} \binom{10}{k} T_k T_{10-k}$

Let $E$ be the ellipse $\frac{x^2}{25} + \frac{y^2}{9} = 1$ and $S(4,0)$ be its focus. A variable tangent $L$ touches $E$ at the point $P(5\cos\theta, 3\sin\theta)$ for $\theta \in [0, 2\pi)$. Let $Q(x,y)$ be the orthogonal projection of $S$ onto $L$. Find the number of values of $\theta$ such that the coordinates of $Q$ satisfy the constraint $|x| + |y| = 7$.

Let $P(x,y)$ be a moving point in the first quadrant on a hyperbola $H$ with its center at the origin and transverse axis along the x-axis. The distance of $P$ to the conjugate axis is $x$. Let $a$ be the semi-transverse axis of $H$. It is given that $\lim_{y \to 0} \frac{x - a - \frac{3}{2}y^2}{y^4} = -\frac{27}{8}$ Find the length of the latus rectum of the hyperbola $H$.

Let $H_e$ be a family of hyperbolas with a fixed transverse axis of length $2$ and eccentricity $e \in (1, \infty)$. Let $L(e)$ denote the length of the conjugate axis of $H_e$. A function $f: (0, \infty) \to \mathbb{R}$ is defined as $f(x) = x^2 - 4x + 3$. Consider the composite function $h(e) = f(L(e))$. If $S$ is the set of all real values of $k$ for which the equation $h(e) = k$ has exactly two distinct solutions for $e \in (1, \infty)$, then the set $S$ is:

Let $E$ be the ellipse $\frac{x^2}{16} + \frac{y^2}{9} = 1$. Let $S$ be the set of all complex numbers $z = x+iy$ lying on the director circle of $E$. For each $z \in S$, let $m_1$ and $m_2$ be the slopes of the two tangents drawn from $z$ to $E$, ordered such that $m_1 > m_2$. Define a complex number $\omega = (y - m_1 x) + i(y - m_2 x)$. If $z_0 \in S$ satisfies $\arg(\omega) = \frac{3\pi}{4}$, find the value of $z_0$.

For a family of ellipses $E_e$ with eccentricity $e \in (0,1)$, the perpendicular distance from the center to the tangent with slope $e$ is given by $p(e) = \sqrt{\frac{1-e}{2+e^2-e^4}}$. Let $R(e)$ be the radius of the director circle of $E_e$. If $f(e) = R(e^2)$, evaluate $\lim_{e \to 1^-} \sqrt{1-e} f'(e)$

Let $S$ be the set of all values of $a^2$ for which the ellipse $E: \frac{x^2}{a^2} + \frac{y^2}{9} = 1$ is tangent to the curve $C: y = 2|x - 3| + 1$. The sum of the elements of $S$ is:

A plane $P$ passes through the point $A(0, 0, 2)$ and is tangent to the ellipsoid $S: \frac{x^2}{3} + \frac{y^2}{2} + z^2 = 1$. If the plane $P$ makes an angle $\theta$ with the plane $x - y = 0$, then the maximum value of $\cos^2 \theta$ is

Let $f(x) = |x^3 - 12x|$. Let $S$ be the set of all $x$-coordinates of the points on the curve $y = f(x)$ at which the tangent line to the curve has a slope $m$, where $m \\in \\{-36, -24, -12, 0, 12, 24, 36\\}$. Find the number of elements in $S$.

A family of conics $C_\lambda$ is defined by the matrix equation $X^T M_\lambda X = 0$, where $X = \begin{pmatrix} x \\ y \\ 1 \end{pmatrix}$ and $M_\lambda = \begin{pmatrix} \lambda & 1 & 0 \\ 1 & \lambda & 0 \\ 0 & 0 & -1 \end{pmatrix}$ The condition that a line $lx + my + n = 0$ is tangent to $C_\lambda$ is given by the vanishing of the bordered determinant: $\Delta(\lambda) = \begin{vmatrix} \lambda & 1 & 0 & l \\ 1 & \lambda & 0 & m \\ 0 & 0 & -1 & n \\ l & m & n & 0 \end{vmatrix} = 0$ A student simplifies $\Delta(\lambda)$ by performing the elementary row operation $R_1 \to \lambda R_1 - R_2$, yielding a new determinant $\Delta'(\lambda)$. The student incorrectly assumes $\Delta'(\lambda) = 0$ is the exact condition of tangency for all $\lambda \in \mathbb{R}$. Consequently, the student finds a specific value $\lambda_0 \in \mathbb{R}$ for which $\Delta'(\lambda_0) = 0$ identically for all $l, m, n$, falsely concluding that every line in the plane is tangent to $C_{\lambda_0}$. Find the equation of the true director circle of the conic $C_{\lambda_0}$.

Let $C_1$ be the conic given by $5x^2 + 6xy + 5y^2 = 20$. Let $P(h,k)$ be a point on the conic $C_2: 5x^2 + 6xy + 5y^2 = \lambda$, where $\lambda > 20$. The tangents drawn from $P$ to $C_1$ touch $C_1$ at points $A(x_1, y_1)$ and $B(x_2, y_2)$. If the absolute value of the determinant $\Delta = \begin{vmatrix} h & k & 1 \\ x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \end{vmatrix}$ is equal to $320$, find the value of $\lambda$.

Let $A, B, C$ be the angles of an acute-angled triangle. A point $P$ is defined as $P(\sin^2 A + \sin^2 B + \sin^2 C, \cos^2 A + \cos^2 B + \cos^2 C)$. Let $L$ be the length of the chord of contact of the tangents drawn from $P$ to the circle $x^2 + y^2 = \frac{9}{2}$. Find the maximum possible value of $L$.

Let $E$ be the ellipse $\frac{x^2}{16} + \frac{y^2}{9} = 1$ and $T$ be the tangent to $E$ at the point $A(0, 3)$. Let $P$ be a finite point in the $xy$-plane such that its chord of contact with respect to $E$ is parallel to $T$. If this chord of contact intersects $E$ in two distinct points, and $d$ is the distance between $T$ and the chord of contact, then the set of all possible values of $d$ is:

A fair coin is tossed. If it lands heads, a point $P(x_0, y_0)$ is chosen uniformly at random from the set $S_1 = \{(x,y) \in \mathbb{Z}^2 \mid \max(|x|,|y|) \le 2\}$. If it lands tails, $P$ is chosen uniformly at random from the set $S_2 = \{(x,y) \in \mathbb{Z}^2 \mid \max(|x|,|y|) = 3\}$. It is observed that a pair of distinct real tangents can be drawn from $P$ to the circle $x^2+y^2=4$, and the slope of their chord of contact is an integer. What is the probability that the coin landed heads?

Let $\Gamma$ be a family of curves in the first quadrant of the $xy$-plane. For any point $P(x,y)$ on a curve in $\Gamma$, consider the conic $C_P$ given by the equation $X^2 + \frac{y}{x} X Y + Y^2 = 1$ in the $XY$-plane. The polar of $P(x,y)$ with respect to $C_P$ is identical to the tangent line to the orthogonal trajectory of $\Gamma$ passing through $P$. If the curve in $\Gamma$ passes through the point $(\sqrt{2}, 1)$, then the value of $y^2$ when $x = 2\sqrt{3}$ is:

Let $C_1, C_2,$ and $C_3$ be three conics given by the equations: $C_1: 2x^2 - 2xy + 2y^2 = 3$ $C_2: x^2 + y^2 = 3$ $C_3: ax^2 + 4xy + by^2 = 3$ For any variable line $L$ not passing through the origin, let its poles with respect to $C_1, C_2,$ and $C_3$ be $P_1, P_2,$ and $P_3$ respectively. If $P_1, P_2,$ and $P_3$ are collinear for all such lines $L$, and $C_3$ is a non-degenerate conic, find the value of $a+b$.

Let $H$ be the hyperbola $\frac{x^2}{9} - \frac{y^2}{16} = 1$. A point $P(t)$ is defined as $P(t) = \left( \frac{3(2t + \cos t)}{e^{2t} - 1}, \frac{4(2t + \cos t)}{e^{2t} - 1} \right)$ for $t > 0$. Let $L_t$ be the polar of $P(t)$ with respect to $H$. As $t \to 0^+$, the line $L_t$ approaches a limiting line $L$. Find the perpendicular distance from the point $A(3, -1)$ to the line $L$.

Let $H$ be a hyperbola whose horizontal and vertical asymptotes are given by the lines $y = 22$ and $x = 22$, respectively. Let $D$ be the diameter of $H$ that bisects a family of parallel chords having slope $m$. The value of $m$ is equal to $\left. \frac{dy}{dx} \right|_{(1,1)}$ for the implicit curve $y^2 - 3y + 2x^{136} = 0$. If the diameter $D$ intersects the $y$-axis at the point $(0, \beta)$, find the value of $\beta$.

Let $E$ be an ellipse in the complex plane given by the equation $|z - 2 - i| + |z - 2 + i| = 4$. If $z_1$ and $z_2$ are the endpoints of any diameter of the ellipse $E$, then the range of the expression $|z_1|^2 + |z_2|^2$ is

Let $A$ be the area of the region in the first quadrant bounded by the ellipse $5x^2 - 6xy + 5y^2 = 8$ and the lines $y = 3x$ and $y = (7+4\sqrt{3})x$. The value of $\frac{36A}{\pi}$ is:

Let $C_t$ be a family of conics given by the equation $\frac{x^2}{f(t)} + \frac{y^2}{g(t)} = 1$, where $t \in (0, 1)$, $f(t) = 2 + t \left[ \frac{3}{t} \right] + t^2 \sin\left(\frac{1}{t}\right)$, and $g(t) = 8 - t \left\{ \frac{3}{t} \right\} + 4t^2 \sin\left(\frac{1}{t}\right)$. Here $[ \cdot ]$ and $\{ \cdot \}$ denote the greatest integer and fractional part functions respectively. Let $D_t$ be the diameter of $C_t$ bisecting a system of parallel chords of slope $m = \frac{1}{t}$. If $M(t)$ is the slope of $D_t$, find the value of $\lim_{t \to 0^+} \frac{M(t) + 2t}{t}$.

Let a family of lines be given by the equation $\Delta(x, y, t) = 0$, where $t$ is a parameter, $a > 0$, and $\Delta(x, y, t) = \begin{vmatrix} x & at^2 & 1 \\\\ y & 2at & t \\\\ 1 & -1 & 0 \end{vmatrix}$ Let $(x_c, y_c)$ be the coordinates of the envelope of this family of lines, which can be obtained by solving the system of equations $\Delta = 0$ and $\frac{\partial \Delta}{\partial t} = 0$. Evaluate the limit: $L = \lim_{t \to 1} \frac{x_c - y_c - a}{t - 1}$

Let $f(x) = \int_0^x (\sqrt{3} + \cos t) dt$. By the Lagrange Mean Value Theorem, there exists at least one point $c \in (0, 2\pi)$ where the tangent line $T$ to the curve $y = f(x)$ is parallel to the secant line joining the points $(0, f(0))$ and $(2\pi, f(2\pi))$. A point $P$ lies on the ellipse $E: x^2 + 3y^2 = 12$ such that the normal to $E$ at $P$ is parallel to $T$. If $P$ lies on the arc of the ellipse where $y$ is a strictly convex function of $x$, find the distance of $P$ from the focus of $E$ having a positive $x$-coordinate.

Let $L$ be the normal line to the parabola $y^2 = 4x$ at the point $P(t^2, 2t)$, where $t > 0$. Let $\vec{u}$ be the vector parallel to $L$ such that $\vec{u} \cdot \hat{i} = 1$. Let $\vec{w}$ be the orthogonal projection of $\vec{u}$ onto the plane $x + y + z = 0$. If the volume of the tetrahedron formed by the origin $O$ and the points with position vectors $\vec{a} = \hat{i} + 4\hat{j} + \hat{k}$, $\vec{b} = 2\hat{i} + \hat{j} + \hat{k}$, and $\vec{w}$ is zero, find the length of the segment $OP$.

Let $H$ be the hyperbola $\frac{x^2}{36} - \frac{y^2}{4} = 1$. For any point $P(x, y)$ on $H$ with $y > 0$, let $A(x)$ be the area of the region bounded by $H$, the x-axis, and the normal to $H$ at $P$. Let $L = \lim_{x \to -\infty} x^2 \left( A(x) - \frac{5}{27} x^2 + 6 \ln|x| + \frac{10}{3} - 6 \ln 3 \right)$ Find the value of $6L + 5$.

A closed curve $C$ is defined by the equation $y^2 + 4|x| = 20$. A light ray originates from the point $M(0, 2)$ and travels parallel to the negative x-axis. It strikes the inner surface of $C$ at point $A$ and undergoes continuous reflections. Let $P$ be the polygon formed by the sequence of all distinct points of incidence on $C$. Find the area of $P$.

A mirror surface $S$ is parameterized by $\vec{r}(u,v) = u^3 \hat{i} + v^3 \hat{j} + (2u^3 - 3v^3 + u^3 v^3) \hat{k}$ for $u, v \in \mathbb{R}$. A light ray with incident direction vector $\vec{v}_i = \hat{i} + 2\hat{j} + 3\hat{k}$ strikes $S$ at the origin. Find the reflected direction vector $\vec{v}_r$ such that $|\vec{v}_r| = |\vec{v}_i|$.

Let $E$ be the ellipse $\frac{x^2}{4} + \frac{y^2}{3} = 1$ with foci $S_1$ and $S_2$, where $S_1$ has a positive abscissa. A ray of light emanating from $S_1$ strikes the ellipse at a point $P$ and is reflected. The line containing the reflected ray intersects the line passing through $S_1$ and perpendicular to the tangent at $P$ at a point $Q$. If the point $P$ is restricted to the region $x \ge 0$ and $y \ge 0$, the locus of $Q$ is a curve of length $L$. Find the value of $L$.

A ray of light originating from the origin $O(0,0)$ travels in the $xy$-plane and strikes the outer surface of a circular mirror whose equation is $x^2 + (y-2)^2 = 1$. After reflection, the ray travels parallel to the x-axis. If the point of incidence $P(x, y)$ lies in the first quadrant, what is the exact value of its y-coordinate?

Let $z_1, z_2 \in \mathbb{C}$ be distinct points lying on the rectangular hyperbola $z^2 + \bar{z}^2 = 4$. If $z_1 + z_2 = 6 + 2i$, find the value of $|z_1 - z_2|^2$.

Let the rectangular hyperbola $H$ be given by the equation $x^2 - y^2 - 4x + 6y - 9 = 0$. Let $f: (-\infty, 0] \to [3, \infty)$ be a bijective function whose graph is a branch of $H$. If $g(x)$ is the inverse function of $f(x)$, find the value of $g'(3+\sqrt{5})$.

Let $M$ be the $3 \times 3$ symmetric matrix representing the rectangular hyperbola $H: x^2 - y^2 + 4xy + 6x - 8y + K = 0$. A new matrix $N$ is obtained by applying the elementary row operation $R_3 \to R_3 + 4R_1 - 2R_2$ to $M$. If the conic represented by the equation $\begin{pmatrix} x & y & 1 \end{pmatrix} N \begin{pmatrix} x \\ y \\ 1 \end{pmatrix} = 0$ degenerates into a pair of straight lines, then the value of $K$ is:

Let $\vec{a} = 3\hat{i} + \hat{j}$, $\vec{b} = \hat{i} + 5\hat{j}$, and $\vec{c} = 14\hat{i} + 18\hat{j}$. A conic section $H$ in the $xy$-plane is defined as the locus of a point $P$ with position vector $\vec{r} = x\hat{i} + y\hat{j}$ satisfying the vector equation: $(\vec{r} \times \vec{a}) \cdot (\vec{b} \times \vec{r}) + 4|\vec{r}|^2 = \vec{r} \cdot \vec{c}$ If $\vec{r}_c$ is the position vector of the center of $H$, find the value of $\vec{r}_c \cdot (2\hat{i} - \hat{j})$.

Find the sum of all integers $k$ such that the conics $C_1: y = x^2 - kx$ and $C_2: x = y^2 - ky$ intersect in exactly four distinct real points with $x$-coordinates $x_1, x_2, x_3, x_4$, and satisfy the inequality $\sum_{i=1}^4 [x_i]^2 \le 53$, where $[ \cdot ]$ denotes the greatest integer function.

A circle $C: x^2 + y^2 - 10x - 2y = 0$ intersects the parabola $y^2 = 4x$ at the origin $O$ and three other distinct points $P, Q, R$. Let $t_1, t_2, t_3$ be the parameters of $P, Q, R$ respectively, where the coordinates of any point on the parabola are given by $(t^2, 2t)$. Let $M = \begin{pmatrix} t_1 & t_2 & t_3 \\ t_2 & t_3 & t_1 \\ t_3 & t_1 & t_2 \end{pmatrix}$. If $B = \text{adj}(M - 2I)$, where $I$ is the $3 \times 3$ identity matrix, find the value of $|B|$.

Let $S$ be the conic given by $x^2 - y^2 - 2x + 4y - 4 = 0$. A variable line $L: lx + my = 1$ intersects $S$ at two distinct points $P$ and $Q$. The position vectors $\vec{p}$ and $\vec{q}$ of $P$ and $Q$ with respect to the origin $O$ are linearly independent, and the scalar projection of $\vec{p}$ along $\vec{q}$ is zero.

As $L$ varies, it envelopes a curve $E$. The altitudes of the triangle formed by any three such lines $L$ are concurrent at a point $H$. As the three lines vary, the locus of $H$ is a straight line $D$. Which of the following points lies on $D$?