Straight Line and a Point

In what ratio does the point $(-4, 6)$ divides the line segment joining the points $A(-6, 10)$and $B(3, -8)$?

The line $3x+2y=24$ meets the $\mathrm{y}$-axis at $\mathrm{A}$ & the $\mathrm{x}$-axis at $\mathrm{B}$. The perpendicular bisector of $\mathrm{AB}$ meets the line through $(0, -1)$ parallel to $\mathrm{x}$-axis at $\mathrm{C}$. Then the area ( in square units ) of the triangle $\mathrm{ABC}$ is

The equation of the perpendicular bisectors of the sides $AB$ and $AC$ of a  $\Delta ABC$ are  $x-y+5=0$  and  $x+2y=0$  respectively. If the point $A$ is  $\left(1,-2\right),$ then the equation of the line $BC$ is.

In a triangle $ABC, D$ and $E$ divide the sides $BC$ and $CA$ in the ratio $2:1$ respectively. If $P$ is the point of intersection of $AD$ and $BE$ then the ratio in which $P$ divides $AD$ is

If $Q$ is the image of the point $P\left(1,1\right)$ with respect to the straight line $x+y+1=0$, then the length of the perpendicular drawn from $Q$ to the line $3x-4y+3=0$ is

If the line $2x-3y+4=0$ divides the line segment joining the points $A\left(-2,3\right)$ and $B\left(3,-2\right)$ in the ratio $m:n$, then the point which divides $AB$ in the ratio $-4 \mathrm{m}:3 \mathrm{n}$ is

Look at the graph and answer the question.

The blue triangle is at point $\left(-4, 5\right)$. What is at $\left(4, -5\right)?$ _____.

Point $P$ divides line $AB$, with $A\left(5,14\right)$ and $B\left(7,16\right)$ in ratio $3:2$ externally. Find the coordinates of $P$.

Point $P$ divides line $AB$, with $A\left(2,3\right)$ and $B\left(5,6\right)$ in ratio $3:2$ externally. Find the coordinates of $P$.

Point $P$ divides line $AB$, with $A\left(7,6\right)$ and $B\left(5,10\right)$ in ratio $3:2$ externally. Find the coordinates of $P$.

Point $P$ divides line $AB$, with $A\left(5,4\right)$ and $B\left(8,6\right)$ in ratio $3:2$ externally. Find the coordinates of $P$.

Point $P$ divides line $AB$, with $A\left(3,4\right)$ and $B\left(7,12\right)$ in ratio $3:2$ externally. Find the coordinates of $P$.

If the equation of a line which divides the line segment joining the points $\left(1,0\right)$ and $\left(3,0\right)$ in the ratio $2:1$ and is also perpendicular to it, is $ax=7$, then the value of $a$ is equal to

If $m$ and $n$ are the lengths of the perpendicular from the origin to the straight lines whose equations are $x\cot \theta -y=2\cos \theta$ and $4x+3y=-\sqrt{5}\cos 2\theta$ $(\theta \in (0,\pi \left)\right),$ respectively, then the value of $m^2+5n^2$ is

If the points $\left(x,-3x\right)$ and $\left(3,4\right)$ lie on the opposite sides of the line $3x-4y=8$, then

Let $S$ be the focus of parabola $x^2+8y=0$ and $Q$ be any point on it. If $P$ divides the line segment $SQ$ in the ratio $1 : 2,$ then the locus of $P$ is

Area of the triangle with vertices $\left(-2,2\right),\left(1,5\right)$ and $\left(6,-1\right)$ is

The distance of the point $\left(3,-5\right)$ from the line $3x-4y-26=0$ is

The area of a triangle is $5$ sq units. Two of its vertices are $\left(2,1\right)$ and $\left(3,-2\right)$$.$ The third vertex lies on $y=x+3$$,$ then the coordinates of the third vertex can be

If the points $\left(2a,a\right),\left(a,2a\right)$ and $\left(a,a\right)$ enclose a triangle of area $18$ sq units, then the centroid of the triangle is equal to