Angle Between Two Straight Lines When Their Equations Are Given

Find the tangent of an angle between the lines $\frac{x}{a}+\frac{y}{b}=1$ and $\frac{x}{a}-\frac{y}{b}=1$.

Show that the line $a^{2}x+ay+1=0$ is perpendicular to the line $x-ay=1$ for all non-zero real values of $a$.

Find the tangent of the angle between the lines which have intercepts $3, 4$ and $1, 8$ on the $x$-axis, and $y$-axis respectively.

Find the angle between the lines $x=a$ and $by+c=0$

If $\theta$ is the angle which the straight line joining the points $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ subtends at the origin, prove that $\tan \theta =\frac{x_2{y}_1-x_1{y}_2}{x_1{x}_2+y_1{y}_2}$ and $\cos \theta =\frac{x_1{x}_2+y_1{y}_2}{\sqrt{x_1^2+y_1^2\sqrt{x_2^2+y_2^2}}}$.

Find the angle between the line joining the points $\left(2,0\right),\left(0,3\right)$ and the line $x+y=1$.

Find the acute angle between the lines $2x-y+3=0$ and $x+y+2=0$.

Find the angle between the following pair of straight lines:

$\left(m^2-mn\right)y=\left(mn+n^2\right)x+n^3$ and $\left(mn+m^2\right)y=\left(mn-n^2\right)x+m^3$

Find the angle between the following pair of straight lines:

$x-4y=3$ and $6x-y=11$.

Find the angles between the following pair of straight lines:

$3x+4y-7=0$ and $4x-3y+5=0$

Find the angle between the following pair of straight lines:

$3x-y+5=0$ and $x-3y+1=0$

Find the angle between the following pair of straight lines:

$3x+y+12=0$ and $x+2y-1=0$

Prove that the straight lines $(a+b) x+(a-b) y=2 ab, (a-b) x+(a+b) y=2ab$ and $x+y=0$ form an isosceles triangle whose vertical angle is $2{\tan }^{-1}\left(\frac{a}{b}\right)$.

Prove that the points$(2,-1), (0,2), (2,3)$ and $(4,0)$ are the coordinates of the vertices of a parallelogram and find the angle between its diagonals.