Dr. SK Goyal Solutions for Chapter: Pair of Straight Lines, Exercise 3: INTRODUCTORY EXERCISE 3.3

If co-ordinate axes are the angle bisectors of the pair of lines $a{x}^2+2hxy+b{y}^2=0$ then:

If the line $y=mx$ is one of the bisector of the lines $x^2+4xy-y^2=0$, then the value of $m=$

The bisectors of the angles between the lines ${\left(ax+by\right)}^2=c{\left(bx-ay\right)}^2,c>0$ are respectively parallel and perpendicular to the line :

If the pairs of straight lines $a{x}^2+2hxy-a{y}^2=0$ and $b{x}^2+2gxy-b{y}^2=0$ be such that each bisects the angles between the other, then prove that $hg+ab=0$.

Prove that the lines $2x^2+6xy+y^2=0$ and $4x^2+18xy+y^2=0$. are equally inclined to the lines $b=1$

Show that the lines bisecting the angle between the bisectors of the angles between the lines $a{x}^2+2hxy+b{y}^2=0$ are given by $\left(a-b\right)\left(x^2-y^2\right)+4hxy=0$.

Prove that the bisectors of the angle between the lines $a{x}^2+acxy+c{y}^2=0$ and $\left(3+\frac{1}{c}\right)x^2+xy+\left(3+\frac{1}{a}\right)y^2=0$ are always the same.

The lines represented by $x^2+2\lambda xy+2y^2=0$ and the lines represented by $\left(1+\lambda \right)x^2-8xy+y^2=0$ are equally inclined, then find the values of $\lambda$.