Straight Lines

Let $\mathbb{R}$ denote the set of all real numbers. Then the area of the region

$$\left\{(x, y) \in \mathbb{R}\times\mathbb{R} : x > 0,\ y > \frac1x,\ 5x - 4y - 1 > 0,\ 4x + 4y - 17 < 0\right\}$$

is

Let $S$ denote the locus of the point of intersection of the pair of lines

$$4x - 3y = 12\alpha,\qquad 4\alpha x + 3\alpha y = 12,$$

where $\alpha$ varies over the set of non-zero real numbers. Let $T$ be the tangent to $S$ passing through the points $(p, 0)$ and $(0, q)$, $q > 0$, and parallel to the line $4x - \dfrac{3}{\sqrt2}y = 0$.

Then the value of $pq$ is

Let $S$ denote the locus of the mid-points of those chords of the parabola $y^2 = x$, such that the area of the region enclosed between the parabola and the chord is $\dfrac43$. Let $\mathcal{R}$ denote the region lying in the first quadrant, enclosed by the parabola $y^2 = x$, the curve $S$, and the lines $x = 1$ and $x = 4$.

Then which of the following statements is (are) TRUE?

Let $P(x_1, y_1)$ and $Q(x_2, y_2)$ be two distinct points on the ellipse $\dfrac{x^2}{9} + \dfrac{y^2}{4} = 1$ such that $y_1 > 0$, and $y_2 > 0$. Let $C$ denote the circle $x^2 + y^2 = 9$, and $M$ be the point $(3, 0)$.

Suppose the line $x = x_1$ intersects $C$ at $R$, and the line $x = x_2$ intersects $C$ at $S$, such that the $y$-coordinates of $R$ and $S$ are positive. Let $\angle ROM = \dfrac{\pi}{6}$ and $\angle SOM = \dfrac{\pi}{3}$, where $O$ denotes the origin $(0, 0)$. Let $|XY|$ denote the length of the line segment $XY$.

Then which of the following statements is (are) TRUE?

Let the straight line $y = 2x$ touch a circle with center $(0, \alpha)$, $\alpha > 0$, and radius $r$ at a point $A_1$. Let $B_1$ be the point on the circle such that the line segment $A_1B_1$ is a diameter of the circle. Let $\alpha + r = 5 + \sqrt{5}$.

Match each entry in List-I to the correct entry in List-II.

The correct option is

Let $A_1, B_1, C_1$ be three points in the $xy$-plane. Suppose that the lines $A_1C_1$ and $B_1C_1$ are tangents to the curve $y^2 = 8x$ at $A_1$ and $B_1$, respectively. If $O = (0,0)$ and $C_1 = (-4, 0)$, then which of the following statements is (are) TRUE?

Let $X = \left\{(x, y) \in \mathbb{Z}\times\mathbb{Z} : \dfrac{x^2}{8} + \dfrac{y^2}{20} < 1 \text{ and } y^2 < 5x\right\}$. Three distinct points $P$, $Q$ and $R$ are randomly chosen from $X$. Then the probability that $P$, $Q$ and $R$ form a triangle whose area is a positive integer, is

Let the position vectors of the points $P, Q, R$ and $S$ be $\vec a = \hat i + 2\hat j - 5\hat k$, $\vec b = 3\hat i + 6\hat j + 3\hat k$, $\vec c = \dfrac{17}{5}\hat i + \dfrac{16}{5}\hat j + 7\hat k$ and $\vec d = 2\hat i + \hat j + \hat k$, respectively. Then which of the following statements is true?

Let $C_1$ be the circle of radius 1 with center at the origin. Let $C_2$ be the circle of radius $r$ with center at the point $A = (4,1)$, where $1 < r < 3$. Two distinct common tangents $PQ$ and $ST$ of $C_1$ and $C_2$ are drawn. The tangent $PQ$ touches $C_1$ at $P$ and $C_2$ at $Q$. The tangent $ST$ touches $C_1$ at $S$ and $C_2$ at $T$. Mid points of the line segments $PQ$ and $ST$ are joined to form a line which meets the $x$-axis at a point $B$. If $AB = \sqrt5$, then the value of $r^2$ is _____.

Let $ABC$ be the triangle with $AB = 1$, $AC = 3$ and $\angle BAC = \dfrac\pi2$. If a circle of radius $r > 0$ touches the sides $AB$, $AC$ and also touches internally the circumcircle of the triangle $ABC$, then the value of $r$ is _____.

Consider the parabola $y^2 = 4x$. Let $S$ be the focus of the parabola. A pair of tangents drawn to the parabola from the point $P = (-2, 1)$ meet the parabola at $P_1$ and $P_2$. Let $Q_1$ and $Q_2$ be points on the lines $SP_1$ and $SP_2$ respectively such that $PQ_1$ is perpendicular to $SP_1$ and $PQ_2$ is perpendicular to $SP_2$. Then, which of the following is/are TRUE?

Consider the hyperbola
$$\frac{x^2}{100} - \frac{y^2}{64} = 1$$
with foci at $S$ and $S_1$, where $S$ lies on the positive $x$-axis. Let $P$ be a point on the hyperbola, in the first quadrant. Let $\angle SPS_1 = \alpha$, with $\alpha < \dfrac\pi2$. The straight line passing through the point $S$ and having the same slope as that of the tangent at $P$ to the hyperbola, intersects the straight line $S_1P$ at $P_1$. Let $\delta$ be the distance of $P$ from the straight line $SP_1$, and $\beta = S_1P$. Then the greatest integer less than or equal to $\dfrac{\beta\delta}{9}\sin\dfrac\alpha2$ is _____.

Let $A\left(\dfrac3{\sqrt a},\sqrt a\right)$, $a>0$, be a fixed point in the $xy$-plane. The image of $A$ in $y$-axis be $B$ and the image of $B$ in $x$-axis be $C$. If $D(3\cos\theta,a\sin\theta)$ is a point in the fourth quadrant such that the maximum area of $\Delta ACD$ is 12 square units, then $a$ is equal to ______.

Let the system of linear equations
$$x+y+\alpha z=2$$
$$3x+y+z=4$$
$$x+2z=1$$
have a unique solution $(x^*,y^*,z^*)$. If $(\alpha,x^*)$, $(y^*,\alpha)$ and $(x^*,-y^*)$ are collinear points, then the sum of absolute values of all possible values of $\alpha$ is

Let the area of the triangle with vertices $A(1,\alpha)$, $B(\alpha,0)$ and $C(0,\alpha)$ be 4 sq. units. If the points $(\alpha,-\alpha)$, $(-\alpha,\alpha)$ and $\left(\alpha^2,\beta\right)$ are collinear, then $\beta$ is equal to

Let a circle $C:(x-h)^2+(y-k)^2=r^2$, $k>0$, touch the $x$-axis at $(1,0)$. If the line $x+y=0$ intersects the circle $C$ at $P$ and $Q$ such that the length of the chord $PQ$ is 2, then the value of $h+k+r$ is equal to ______.

Let $\mathrm P_1$ be a parabola with vertex $(3,2)$ and focus $(4,4)$ and $\mathrm P_2$ be its mirror image with respect to the line $x+2y=6$. Then the directrix of $\mathrm P_2$ is $x+2y=$ ______.

Let a circle C touch the lines $L_1:4x-3y+K_1=0$ and $L_2:4x-3y+K_2=0$, $K_1,K_2\in\mathbb R$. If a line passing through the centre of the circle C intersects $L_1$ at $(-1,2)$ and $L_2$ at $(3,-6)$, then the equation of the circle C is :

Let $x=2t$, $y=\dfrac{t^2}3$ be a conic. Let S be the focus and B be the point on the axis of the conic such that $SA\perp BA$, where A is any point on the conic. If k is the ordinate of the centroid of the $\Delta SAB$, then $\displaystyle\lim_{t\to1}k$ is equal to :

A circle touches both the $y$-axis and the line $x+y=0$. Then the locus of its center is: