Different Forms of the Equation of a Straight Line

Write an equation representing a pair of lines through the point $(a,b)$ and parallel to the coordinate axes.

If $a, b, c$ are in G.P. write the area of the triangle formed by the line $ax+by+c=0$ with the coordinates axes.

If $a\ne b\ne c$, write the condition for which the equations $(b-c) x+(c-a) y+(a-b)=0$ and $\left(b^3-c^3\right)x+\left(c^3-a^3\right)y+\left(a^3-b^3\right)=0$ represent the same line.

Write the integral values of $m$ for which the $x$ -coordinate of the point of intersection of the lines $y=mx+1$ and $3x+4y=9$ is an integer.

Write the sum of the coordinates of the image of the point $(3,8)$ in the line $x+3y-7=0$.

If the diagonals of the quadrilateral formed by the lines $l_{1}x+m_{1}y+n_1=0, l_{2}x+m_{2}y+n_2=0, l_{1}x+m_{1}y+n_1^{\text{'}}=0$ and $l_{2}x+m_{2}y+n_2^{\text{'}}=0$ are perpendicular, then write the value of $l_1^2-l_2^2+m_1^2-m_2^2$.

Write the area of the triangle formed by the coordinate axes and the line $(\sec \theta -\tan \theta )x+(\sec \theta +\tan \theta )y=2$.

Write the value of $\theta \in (0 , \frac{\pi }{2})$ for which area of the triangle formed by points $O(0,0), A(a \cos \theta , b \sin \theta )$ and $B(a \cos \theta ,-b \sin \theta )$ is maximum.

If the centroid of a triangle formed by the points $(0,0), (\cos \theta ,\sin \theta )$ and $(\sin \theta ,-\cos \theta )$ lies on the line $y=2x$, then write the value of $\tan \theta$.

Write the coordinates of the orthocentre of the triangle formed by the lines $x^2-y^2=0$ and $x+6y=18$.

The ratio in which the line $3x+4y+2=0$ divides the distance between the lines $3x+4y+5=0$ and $3x+4y-5=0$ is

The equation of the line with slope $\frac{-3}{2}$ and which is concurrent with the lines $4x+3y-7=0$ and $8x+5y-1=0$ is

The angle between the lines $2x-y+3=0$ and $x+2y+3=0$ is

The centroid of a triangle is $(2,7)$and two of its vertices are $(4,8)$and $(-2,6)$.The third vertex is

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

Find the equation of the straight line which cuts off intercepts on $x-$axis twice that on $y-$axis and is at $a$ unit distance from the origin.

The line $2x+3y=12$ meets the$x-$axis at $A$ and $y-$axis at $B$. The line through $(5,5)$ perpendicular to $AB$ meets the $x-$axis and the line $AB$ at $C$ and $E$ respectively. If $O$ is the origin, find the area of figure $OCEB$.

A line is such that its segment between the straight lines $5x - y - 4 = 0$ and $3x + 4y - 4 = 0$ is bisected at the point $(1, 5).$ Obtain its equation.

Find the distance of the line $2x + y = 3$ from the point $(-1, -3)$ in the direction of the line whose slope is $1.$

Find the distance of the point $(2, 5)$ from the line $3x + y + 4 = 0$ measured parallel to the line $3x - 4y + 8 = 0.$