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.

Let $A\left(4,2\right)$ and $B\left(2,4\right)$ be two given point and $L$ be the straight line $3x+2y+10=0$

Column $1$ indicates different points on line.

Column $2$ indicates different quantity required

Column $3$ indicates value of different points for the required quantity.

Which of the following options is the only CORRECT combination?

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

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

$Q$ is any point on the line $x-a=0$ and $O$ is the origin. If $A$ is the point $(a, 0)$ and $QR$, the bisector of $\mathit{\angle }OQA$, meets the $x$-axis in $R$, show that the locus of the foot of the perpendicular from $R$ to $OQ$ is the curve $(x-2a)\left(x^2+y^2\right)+a^{2}x=0$.

Show that the feet of the perpendiculars drawn from the origin upon the straight lines $x+y-4=0, x+5 y=26$ and $3x-y=10$ are collinear. Find the equation of the line of collinearity.

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

If the image of point $\left(4,-3\right)$ with respect to the line $y=x$ is $\left(a,b\right)$, then find $a+b$.

For the following shaded region, the linear constraints are 

If one of the lines $2x^2-xy+b{y}^2=0$ passes through the point $\left(-4,-2\right),$ then $b^2=$

The equation of the image of line $y=x$ with respect to the line mirror $2x-y=1$ is

Complete the following table :

Complete the following table :

If in $\Delta ABC,AC=3,BC=4$ medians $AD$ and $BE$ are perpendicular then area of $\Delta ABC\_\_\_\_\mathrm{sq}.\mathrm{units}$ 

Image of point $\left(5,2,-7\right)$ in $xy-$plane is given by $\left(x,y,z\right)$. Find product $xyz$.

Find the values of $\alpha$ such that the variable point $(\alpha ,\tan \alpha )$ lies inside the triangle whose sides are $y=x+\sqrt{3}-\frac{\pi }{3}, x+y+\frac{1}{\sqrt{3}}+\frac{\pi }{6}=0$ and $x-\frac{\pi }{2}=0$.

Find the range of $\alpha$ for which the points $(\alpha ,2+\alpha )$ and $\left(\frac{3\alpha }{2},{\alpha }^2\right)$ lie on opposite sides of the line $2 x+3 y=6$.

How the following pairs of points are placed w.r.t. the line $3x-8y-7=0?$
$(-1,-1)$ and $(3,7)$