Distance of a Point From a Line

The length of perpendicular dropped from the origin to the line $3x+ky=10$ is 2 units then values of k is/are

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 ______________

Prove that the area made by $\mathrm{y}={\mathrm{m}}_1\mathrm{x}+{\mathrm{c}}_1, \mathrm{y}={\mathrm{m}}_2\mathrm{x}+{\mathrm{c}}_2 \mathrm{and} \mathrm{x}=0$ is $\frac{1}{2}\frac{{\left({\mathrm{c}}_1-{\mathrm{c}}_2\right)}^2}{\left|{\mathrm{m}}_1-{\mathrm{m}}_2\right|}$.

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

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

The perpendicular distance between the lines $2x-y=11$ and $6x-3y+11=0$ is

Ravi is doing experiment with distance and other parameters between points $\mathrm{P}(3,4)$ and line $\mathrm{L}\equiv x+y=1$. Perpendicular distance between $\mathrm{P}$ and $\mathrm{L}$ is

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

The co-ordinates of the point where line $\frac{x}{a}+\frac{y}{b}=7$ intersects $y-\mathrm{axis}$ are

If the line $x-y+1=0$ cuts the lines $2x+2y+3=0$ and $3x+3y+2=0$ at the points $A$ and $B$ respectively, then $AB=$

The Locus of centers of the circles, possessing the same area and having $3x-4y+4=0$ and $6x-8y-7=0$ as their common tangent, is

The number of lines that can be drawn from the point $\left(2,3\right)$, so that its distance from $\left(-1,6\right)$ is equal to $6$ is

A line makes an intercept of $5 \text{and} 4$ on the $x-$axis and $y-$axis respectively. If the axes are rotated about the origin by an angle $\alpha$ such that the length of the intercept on the new $x-$axis is $8$. The length of intercept on new $y-$axis is

Find the distance between the parallel lines $3x-4y+7=0$ and $3x-4y+5=0$.

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.

Consider two lines $L_1$ and $L_2$ given by $3x+4y-7=0$ and $4x-3y-1=0$ respectively and a variable point $P$. Let $d\left(P,L_i\right), i=1,2$ represents the perpendicular distance of point $P$ from the line $L_{\mathbf{i}}$. If point $P$ moves in a certain region $R$ in such a way that $6\le 2d\left(P,L_1\right)+3d\left(P,L_2\right)\le 12$, then the area of the region $R$ :

Find the distance between the point $\left(1,3\right)$ and the origin.

If $p$ is the length of the perpendicular from the origin on the line $\frac{x}{a}+\frac{y}{b}=1$, then

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

What is the distance of point $\left(5, 12\right)$ from the origin?