General Equation of a Line

Equation of the line passing through $(1,2)$ and parallel to the line $y=3x-1$ is

Find the shortest distance from a hotel located at $(2,4)$ to a straight road with equation $3x+5y+8=0$.

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

Prakash draws a $∆\mathrm{PQR}$ on a whiteboard. The equation of the perpendicular bisectors of its sides $\mathrm{PQ}$ and $\mathrm{PR}$ are $2\mathrm{x}-3\mathrm{y}+6=0$ and $\mathrm{x}+3\mathrm{y}=0$ respectively. What is the equation of line $\mathrm{QR}$ if the point $\mathrm{P}$ is $(2, -3)$?

Chords $\mathrm{AB}$ and $\mathrm{CD}$ of a circle when extended to meet at point $\mathrm{X}.$ Given $\mathrm{AB}=4\mathrm{cm},$ $\mathrm{BX}=6\mathrm{cm}$ and $\mathrm{XD}=5\mathrm{cm},$ calculate the length of $\mathrm{CD}.$ (in $\mathrm{cm}$)

$P\text{'} \mathrm{and} Q\text{'}$be the images of $P\left(-3, 5\right) \mathrm{and} Q\left(-5, 3\right)$ respectively on reflection in $Y- \mathrm{axis}$. Find the coordinates of $P\text{'} \mathrm{and} Q\text{'}$.

In the figure below, lines $K$ and $I$ are parallel. The value of $\mathrm{a}°+\mathrm{b}°$ is -

If the equation of the straight lines passing through the origin and making an angle of $45°$ with the straight line $\sqrt{3}x+y=11$ is $y=(\sqrt{m}\pm 2)x$,

Then find $m.$

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

Two lines intersect

If the sum of the distances of a moving point in a plane from the axes is $1$, then find the locus of the point.

[Hint: Given that $\left|x\right|+\left|y\right|=1,$ which gives four sides of a square.]

If the intercept of a line between the coordinate axes is divided by the point $(-5,4)$ in the ratio $1:2,$ then find the equation of the line.

The equation of lines passing through $(1,2)$ and making angle $30°$ with $y$-axis is$y-\sqrt{a}x+\sqrt{b}-c=0$. Find $a+b+c.$

$\overrightarrow{\mathit{A}\mathit{B}}\mathit{=}\mathit{3}\hat{\mathit{i}}\mathit{-}\hat{\mathit{j}}\mathit{+}\hat{\mathit{k}}$ and $\overrightarrow{\mathit{C}\mathit{D}}\mathit{=}\mathit{-}\mathit{3}\hat{\mathit{i}}\mathit{+}\mathit{2}\hat{\mathit{j}}\mathit{+}\mathit{4}\hat{\mathit{k}}$ are two vectors. The position vectors of the points $A$ and $C$ are $6i+7\hat{j}+4\hat{k}$ and $-9\hat{j}+2\hat{k}$, respectively. Find the position vector of a point $P$ on the line $AB$ and a point $Q$on the line $CD$ such that $\overline{\mathit{P}\mathit{Q}}$ is perpendicular to $\overline{\mathit{A}\mathit{B}}$ and $\overline{\mathit{C}\mathit{D}}$ both.

A ray $\left(L_1\right)$ is incident on  the mirror $\left(L_2\right)$ $3 \mathrm{x}+4 \mathrm{y}=5$ and the equation of reflected ray $\left(L_3\right)$ is $\mathrm{x}+\mathrm{y}=1,$ the equation of$\left(L_1\right)$  is

The slope of a line is double of the slope of another line. If tangent of the angle between them is $\frac{1}{3},$ find the slopes
of he lines.

If $16a^2+25b^2-9c^2+40ab=0,$ then family of straight lines $ax+by+c=0$ will be concurrent at:

Find the value of $p$ so that the three lines $3x+y-2=0,px+2y-3=0$ and $2x-y-3=0$ may intersect at one point.

The distance of the point $\left(3,5\right)$ from $2x+3y-14=0$ measured parallel to $x-2y=1$ is