Straight Line and a Point

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.

Consider the triangles with vertices $A\left(2, 1\right), B\left(0, 0\right)$ and $C\left(t, 4\right), t=\left[0, 4\right]$. If the maximum and the minimum perimeters of such triangles are obtained at $t=\alpha$ and $t=\beta$ respectively, then $6\alpha +21\beta$ is equal to ___________.

Let the equations of two adjacent sides of a parallelogram $ABCD$ be $2x-3y=-23$ and $5x+4y=23$. If the equation of its one diagonal $AC$ is $3x+7y=23$  and the distance of $A$ from the other diagonal is $d$, then $50d^2$ is equal to ______________

A point $P\left(x,y\right)$ moves such that the sum of its distances from the lines $2x-y-3=0$ and $x+3y+4=0$ is $7$, then the area bounded by locus of $P$ is

Which of the following point lie inside the triangle formed by $x+y=5$ and the coordinate axes?

The refection of $y=\sqrt{x}$ w.r.t. $y-$axis is

If $L_1,L_2$ denotes the line $x+2y-2=0$, $2x+3y+4=0$, then

Foot of the perpendicular of $\left(6,8\right)$ on the line $x=y$ is 

The number of points $p\left(x,y\right)$ with natural numbers as coordinates that lie inside the quadrilateral formed by the lines $2x+y=2, x=0, y=0$ and $x+y=5$ is 

If the points $A\left(-5, 3\right), B\left(-1, -1\right), C\left(3, 2\right) \& D\left(-1, 6\right)$ are the vertices of a parallelogram taken in order, then taking $BC$ as the base, find the height of the parallelogram.

In straight lines, prove that the reflection of $\left(x, y\right)$ with respect of $x-$axis is $\left(x, -y\right)$ using the foot of perpendicular method

Two mutually perpendicular straight lines through the origin form an isosceles triangle with the line $2x+y=5$, then the area (in sq. units) of the triangle is

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

If $a{x}^2-b{y}^2+cxy-50=0$ is the reflection of $xy=4$ in the line $y=2x$, then the value of $\left(\frac{107}{c}-\frac{7a}{b}\right)$ is

The perpendicular distance of the straight line $3x+4y=5\sqrt{5}$ from the origin is ______

If $L=0$ is the equation of reflected ray of a ray of light along $x+\sqrt{3}y-\sqrt{3}=0$ about $x$- axis, then the perpendicular distance of point $\left(2,-1\right)$ from $L$ is

Let $S$ be a subset of the plane defined by $S=\left\{\left(x,y\right):\left|x\right|+2\left|y\right|=1\right\}$. Then, the radius of the smallest circle with centre at the origin and having non-empty intersection with $S$ is

For the following shaded region, the linear constraints are