Various Forms of the Equation of a Line

The equations of the normal to the curve  $x=1-\cos \theta :y=\theta -\sin \theta$ at $\theta =\frac{\pi }{4}$  would be:

A straight line in drawn through the centre of a square $ABCD$ intersecting side $AB$ at the point $N$ so that $AN:NB=1:2$ On this line take an arbitrary point $M$ lying inside the square. Prove that the distances from $M$ to the sides of the square $AB, AD, BC$ and $CD$ taken in that order form an $A.P.$

Find the equation of the internal bisectors of the angle $BAC$ of the triangle $ABC$, where the coordinates of the vertices $A, B$ and $C$ are respectively $(3, 5), (-2, 2)$ and $(5, 1)$.

Determine which of the angles (acute or obtuse) formed by the lines $x-7y+2=0$ and $5x+5y-4=0$ contains the point $(-1, 3)$.

Find the equation of the straight line which cuts intercepts $4$ and $-3$ units from the axes.

One side of a square is inclined to the $x$-axis at an angle $\varphi$ and one of its extrimities is at the origin. Prove that the equations of its diagonals are $y(\cos \varphi -\sin \varphi )=x(\cos \varphi +\sin \varphi )$ and $y(\sin \varphi +\cos \varphi )=x(\cos \varphi -\sin \varphi )$.

Write down the equation of the straight line passing through $\left(2, 3\right)$ and is parallel to the line $4x-3y=5$

Find the area of the triangle formed by the axes of co-ordinates and the straight line $3x+4y=12$.

Prove that the equation of the chord joining the points$(a \sec \theta , b \tan \theta )$ and $(a \sec \varphi , b \tan \varphi )$ on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is $\frac{x}{a}\cos \frac{\theta -\varphi }{2}-\frac{y}{b}\sin \frac{\theta +\varphi }{2}=\cos \frac{\theta +\varphi }{2}$.

Show that the equation of the chord joining the points $(a \cos \theta , b \sin \theta )$ and $(a \cos \varphi , b \sin \varphi )$ on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is $\frac{x}{a}\cos \frac{\theta +\varphi }{2}+\frac{y}{b}\sin \frac{\theta +\varphi }{2}=\cos \frac{\theta -\varphi }{2}$.

Find the equation of the straight line passing through the points $(a \cos \alpha , b \sin \alpha )$ and $(a \cos \beta , b \sin \beta )$. Hence show that, the straight line passes through the origin when $|\alpha -\beta |=\pi$.

One diagonal of a square is the portion of the line $\frac{x}{a}+\frac{y}{b}=1$ intercepted between the axes. Show that the extremities of the other diagonal are$\left(\frac{a+b}{2}, \frac{a+b}{2}\right)$ and $\left(\frac{a-b}{2}, \frac{b-a}{2}\right)$.

The equation of the perpendicular bisectors of the sides $AB$ and $AC$ of triangle $ABC$ are $x-y+5=0$ and $x+2y=0$ respectively. If the co-ordinates of the point $A$ are $(1, -2)$ find the equation of the line $BC$.

A straight line $L$ is perpendicular to the line $5x-y=1$. The area of the triangle formed by the line $L$ with coordinate axis is $5 \mathrm{sq}.$units. Find the equation of the line $L$.

A vertex of an equilateral triangle is at $(2, 3)$ and the equation of the opposite side is $x+y=2$. Find the equation of the other two sides.

Find the equations of the straight lines which pass through the point of intersection of the lines $21x+8y=18$ and $11x+3y+12=0$ and each of them makes a triangle with the co-ordinate axes of area $9$ square unit.

Find the equation of the straight line passing through the point of intersection of the straight lines $2x+3y+4=0$ and $3x+y-1=0$ and inclined to the positive direction of the $x$-axis at an angle $135°$.