R D Sharma Solutions for Chapter: The Straight Lines, Exercise 10: EXERCISE 23.10

Find the equation of the line passing through the point of intersection of the lines $4x - 7y - 3 = 0$ and $2x - 3y + 1 = 0$ that has equal intercepts on the axes.

Show that the area of the triangle formed by the lines $y=m_{1}x, y=m_{2}x$ and $y=c$ is equal to $\frac{c^2}{4}(\sqrt{33}+\sqrt{11})$, where $m_1, m_2$ are the roots of the equation $x^2+(\sqrt{3}+2)x+\sqrt{3}-1=0$.

Find the orthocentre of the triangle the equations of whose sides are $x+y=1, 2x+3y=6$ and $4x-y+4=0$.

Three sides $AB, BC$ and $CA$ of a triangle $ABC$ are $5x-3y+2=0, x-3y-2=0$ and $x+y-6=0$ respectively. Find the equation of the altitude through the vertex $A$.

Find the coordinates of the orthocentre of the triangle whose vertices are $(-1,3), (2,-1)$ and $(0,0)$.

Find the coordinates of the incentre and centroid of the triangle whose sides have the equations $3x-4y=0,12y+5x=0$ and $y-15=0$.

Prove that the lines $\sqrt{3}x+y=0, \sqrt{3}y+x=0, \sqrt{3}x+y=1$ and $\sqrt{3}y+x=1$ form a rhombus.

Find the equation of the line passing through the intersection of the lines $2x+y=5$ and $x+3y+8=0$ and parallel to the line $3x+4y=7$.