Andhra Pradesh Board Solutions for Chapter: The Straight Line, Exercise 5: Exercise 3(e)

Find the circumcenter of the triangle whose sides are given by $x+y+2=0, 5x-y-2=0$ and $x-2y+5=0$.

Find the equations of the straight lines passing through $(1,1)$ and which are at a distance of $3$ units from $(-2,3)$.

If $p$ and $q$ are the lengths of the perpendiculars from the origin to the straight lines $x\sec \alpha +ycosec\alpha =a$ and $x\cos \alpha -y\sin \alpha =a\cos 2\alpha ,$ prove that $4p^2+q^2=a^2$.

Two adjacent sides of a parallelogram are given by $4x+5y=0$ and $7x+2y=0$ and one diagonal is $11x+7y=9$. Find the equations of the remaining sides and the other diagonal.

Find the incentre of the triangle formed by the straight lines $x+1=0, 3x-4y=5$ and $5x+12y=27$.

Find the incentre of the triangle formed by the straight lines $x+y-7=0, x-y+1=0$ and $x-3y+5=0$.

A triangle is formed by the lines $ax+by+c=0, lx+my+n=0$ and $px+qy+r=0$. Given that the triangle is not right-angled, show that the straight line $\frac{ax+by+c}{ap+bq}=\frac{lx+my+n}{lp+mq}$ passes through the orthocenter of the triangle.

The Cartesian equations of the sides $BC\mathit{,}\mathit{ }CA$ and $AB$ of a triangle are respectively $u_r\equiv a_{r}x+b_{r}y+c_r=0, r=1, 2, 3$. Show that the equation of the straight line passing through $A$ and bisecting the side $BC$ is $\frac{u_3}{a_3{b}_1-a_1{b}_3}=\frac{u_2}{a_1{b}_2-a_2{b}_1}$.