Dr. SK Goyal Solutions for Chapter: The Straight Lines, Exercise 8: EXERCISE ON LEVEL-II

Given the pencil of lines $a\left(2x+5y+4\right)+b\left(3x-2y+25\right)=0$. Find the line of pencil from which the point $M\left(4,5\right)$ is at the greatest distance.

A triangle has the lines $y=m_{1}x$ and $y=m_{2}x$ as two of its sides, with $m_1$ and $m_2$ being roots of the equation $b{x}^2+2hx+a=0$. If $H\left(a,b\right)$ is the orthocentre of the triangle, show that the equation of the third side is
$\left(a+b\right)\left(ax+by\right)=a b\left(a+b-2h\right)$

The base of a triangle is fixed and the difference of the base of angles is given. Find the locus of the vertex, if $\alpha$ is the difference angle.

The base $BC$ of a triangle $ABC$ is bisected at the point $\left(a,b\right)$ and the equations to the sides $AB$ and$AC$ are $ax+by=1$ and $bx+ay=1$ respectively. Show that the equation to the median through $A$ is
$\left(2ab-1\right)\left(ax+by-1\right)=\left(a^2+b^2-1\right)\left(bx+ay-1\right)$.

The equations of the sides of a triangle are $L_i\equiv a_{i}x+b_{i}y+c_i=0; \left(i=1,2,3\right)$. Show that the co-ordinates of the orthocentre of the triangle satisfy the equations
${\lambda }_1{L}_1={\lambda }_2{L}_2={\lambda }_3{L}_3$, where ${\lambda }_1=a_2{a}_3+b_2{b}_3; {\lambda }_2=a_3{a}_1+b_3{b}_1; {\lambda }_3=a_1{a}_2+b_1{b}_2$.

A ray of light sent along the straight line $y=\frac{2x}{3}-4$. On reaching the $x$-axis it is reflected. Find the point of incidence and the equation of the reflected ray.

The co-ordinates of the extremities $A$ and $B$ of a rod are $\left(1,2\right)$ and $\left(3,4\right)$ respectively. $S\left(0,0\right)$ is a point of source of light. The rod $AB$ is parallel to the wall and is at equal distance from $S$ and the wall. If $CD$ is the shadow of $AB$ on the wall. $S,AB$ and $CD$ are coplanar. Find the co-ordinates of $C$ and $D$ and the length $CD$.

$Q$ is any point on the line $x-a=0$. If $A$ is the point $\left(a,0\right)$ and $QR$, the bisector of the angle $OQA$, meets the $x$-axis in $R$, prove that the locus of foot of the perpendicular from $R$ to $OQ$ is $\left(x-a\right)\sqrt{\left(x^2+y^2\right)}=a\left|y\right|$.