Equations of a line

Let the vertex A of a triangle ABC be $(1,2)$, and the mid-point of the side AB be $(5,-1)$. If the centroid of this triangle is $(3,4)$ and its circumcenter is $(\alpha,\beta)$, then $21(\alpha+\beta)$ is equal to:

From the point $(-1,-1)$, two rays are sent making angles of 45° with the line $x+y=0$. These rays get reflected from the mirror $x+2y=1$. If the equations of the reflected rays are $ax+by=9$ and $cx+dy=7$, $a,b,c,d\in\mathbf Z$, then the value of $ad+bc$ is ________.

Let the line $x-y=4$ intersect the circle $C:(x-4)^2+(y+3)^2=9$ at the points Q and R. If $P(\alpha,\beta)$ is a point on C such that $PQ=PR$, then $(6\alpha+8\beta)^2$ is equal to ________.

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 $P(x_1, y_1)$ 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 $A_1, B_1, C_1$ 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 $\mathrm P_1$ 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 C be a circle passing through the points A$(2,-1)$ and B$(3,4)$. The line segment AB is not a diameter of C. If r is the radius of C and its centre lies on the circle $(x-5)^2+(y-1)^2=\dfrac{13}2$, then $r^2$ is equal to :

In an isosceles triangle ABC, the vertex A is $(6,1)$ and the equation of the base BC is $2x+y=4$. Let the point B lie on the line $x+3y=7$. If $(\alpha,\beta)$ is the centroid of $\Delta ABC$, then $15(\alpha+\beta)$ is equal to :

A ray of light passing through the point $P(2,3)$ reflects on the $x$-axis at point A and the reflected ray passes through the point $Q(5,4)$. Let R be the point that divides the line segment AQ internally into the ratio 2:1. Let the co-ordinates of the foot of the perpendicular M from R on the bisector of the angle PAQ be $(\alpha,\beta)$. Then, the value of $7\alpha+3\beta$ is equal to ______.

Let a triangle be bounded by the lines $L_1:2x+5y=10$; $L_2:-4x+3y=12$ and the line $L_3$, which passes through the point $P(2,3)$, intersects $L_2$ at A and $L_1$ at B. If the point P divides the line-segment AB, internally in the ratio $1:3$, then the area of the triangle is equal to :

The distance of the origin from the centroid of the triangle whose two sides have the equations $x-2y+1=0$ and $2x-y-1=0$ and whose orthocenter is $\left(\dfrac73,\dfrac73\right)$ is:

If $xy^4$ attains maximum value at the point $(x,y)$ on the line passing through the points $(50+\alpha,0)$ and $(0,50+\alpha)$, $\alpha>0$, then $(x,y)$ also lies on the line :

Let $\alpha_1,\alpha_2$ $(\alpha_1<\alpha_2)$ be the values of $\alpha$ for the points $(\alpha,-3)$, $(2,0)$ and $(1,\alpha)$ to be collinear. Then the equation of the line, passing through $(\alpha_1,\alpha_2)$ and making an angle of $\dfrac\pi3$ with the positive direction of the $x$-axis, is :

A line, with the slope greater than one, passes through the point $A(4,3)$ and intersects the line $x-y-2=0$ at the point B. If the length of the line segment AB is $\dfrac{\sqrt{29}}3$, then B also lies on the line :

The equations of the sides AB, BC and CA of a triangle ABC are $2x+y=0$, $x+py=15a$ and $x-y=3$ respectively. If its orthocentre is $(2,a)$, $-\dfrac12<a<2$, then p is equal to ______.

Let A(1, 1), B(−4, 3), C(−2, −5) be vertices of a triangle ABC, P be a point on side BC, and $\Delta_1$ and $\Delta_2$ be the areas of triangles APB and ABC, respectively. If $\Delta_1:\Delta_2=4:7$, then the area enclosed by the lines AP, AC and the $x$-axis is

The equations of the sides AB, BC and CA of a triangle ABC are $2x+y=0$, $x+py=39$ and $x-y=3$ respectively and P(2, 3) is its circumcentre. Then which of the following is NOT true ?

Let the circumcentre of a triangle with vertices A(a, 3), B(b, 5) and C(a, b), ab > 0 be P(1, 1). If the line AP intersects the line BC at the point Q(k$_1$, k$_2$), then k$_1$ + k$_2$ is equal to :

If the system of linear equations

$\lambda x+y-z=-1$
$x-y-3z=2$
$-x+y+z=\mu$

has infinitely many solutions, then the equation of the line passing through the points $(\lambda+2\mu,\ 2\lambda+\mu)$ and $(1,\ \lambda\mu)$ is