Distance and section formulae

Let the mid points of the sides of a triangle $ABC$ be $\left(\frac52,7\right)$, $\left(\frac52,3\right)$ and $(4,5)$. If its incentre is $(h,k)$, then $3h+k$ is equal to :

Let O be the origin, and P and Q be two points on the rectangular hyperbola $xy=12$ such that the mid point of the line segment PQ is $\left(\dfrac12,-\dfrac12\right)$. Then the area of the triangle OPQ equals :

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:

In an equilateral triangle PQR, let the vertex P be at $(3,5)$ and the side QR be along the line $x+y=4$. If the orthocentre of the triangle PQR is $(\alpha,\beta)$, then $9(\alpha+\beta)$ is equal to:

For any two points $M$ and $N$ in the $XY$-plane, let $\overrightarrow{MN}$ denote the vector from $M$ to $N$, and $\vec 0$ denote the zero vector. Let $P$, $Q$ and $R$ be three distinct points in the $XY$-plane. Let $S$ be a point inside the triangle $\Delta PQR$ such that

$$\overrightarrow{SP} + 5\,\overrightarrow{SQ} + 6\,\overrightarrow{SR} = \vec 0.$$

Let $E$ and $F$ be the mid-points of the sides $PR$ and $QR$, respectively. Then the value of

$$\frac{\text{length of the line segment } EF}{\text{length of the line segment } ES}$$

is _____.

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 _____.

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 $x=2t$, $y=\dfrac{t^2}3$ 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 :

The line $y=x+1$ meets the ellipse $\dfrac{x^2}4+\dfrac{y^2}2=1$ at two points $P$ and $Q$. If $r$ is the radius of the circle with $PQ$ as diameter then $(3r)^2$ 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 rectangle R with end points of one of its sides as $(1,2)$ and $(3,6)$ is inscribed in a circle. If the equation of a diameter of the circle is $2x-y+4=0$, then the area of R is ______.

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 $l$ be a line which is normal to the curve $y=2x^2+x+2$ at a point P on the curve. If the point $Q(6,4)$ lies on the line $l$ and O is origin, then the area of the triangle OPQ 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: