Dr. SK Goyal Solutions for Chapter: The Straight Lines, Exercise 3: INTRODUCTORY EXERCISE 2.3

If $a,b,c$ are in $\mathrm{AP}$ then $ax+by+c=0$ represents :

Find the equation of the line through the intersection of $2x-3y+4=0$ and $3x+4y-5=0$ and perpendicular to $6x-7y+c=0$:

The locus of the point of intersection of the lines $\frac{x}{a}-\frac{y}{b}=m,\frac{x}{a}+\frac{y}{b}=\frac{1}{m}$ is :

If the lines $\left(a-b-c\right)x+2ay+2a=0, 2bx+\left(b-c-a\right)y+2b=0$ and $\left(2c+1\right) x+2 c y+c-a-b=0$ are concurrent, then prove that either $a+b+c=0$ or ${\left(a+b+c\right)}^2+2a=0$.

Prove that the lines $ax+by+c=0, bx+cy+a=0$ and $cx+ay+b=0$ are concurrent if $a+b+c=0$ or $a+b\omega +c{\omega }^2=0$ or $a+b{\omega }^2+c\omega =0$, where $\omega$ is a complex cube root of unity.

Find the equation of the straight line which passes through the intersection of the lines $x-y-1=0$ and $2x-3y+1=0$ and is parallel to $3x+4y=14$.

Let $a,b,c$ be parameters. Then, the equation $ax+by+c=0$ will represent a family of straight lines passing through a fixed-point, if there exists a linear relation between $a,b$ and $c$.

Prove that $\left(-a,-\frac{a}{2}\right)$ is the orthocentre of the triangle formed by the lines $y=m_{i}x+\frac{a}{m_i},i=1,2,3; m_1,m_2,m_3$ being the roots of the equation $x^3-3x^2+2=0.$