91Ó°ÊÓ

A variable circle always touches the line \(y=x\) and passes through the point \((0,0)\). Show that the common chords of this circle and \(x^{2}+y^{2}+6 x+8 y-7=0\) will pass through a fixed point \(\left(\frac{1}{2}, \frac{1}{2}\right)\).

Show that the locus of the points of chords of contact of tangents subtending a right angle at the centre is a concentric circle whose radius is \(\sqrt{2}\) times the radius of the given circle. Also show that this is also the locus of the point of intersection of perpendicular tangents.

Three sides of a triangle have the equations \(L_{i}=y-m_{r} x-c_{r}=0, r=1,2,3\). Then show that \(\lambda L_{2} L_{3}+\mu L_{3} L_{1}+v L_{1} L_{2}=0\) where \(\lambda, \mu, v \neq 0\) is the equation of the circumcircle of the triangle if \(\sum \lambda\left(m_{2}+m_{3}\right)=0\) and \(\sum \lambda\left(m_{2} m_{3}-m_{1}\right)=0\).

A triangle is formed by the lines whose combined equation is \(c(x+y-4)(x y-2 x-y+2)=0\). Show that the equation of its circumference is \(x^{2}+y^{2}-3 x-5 y+8=0\).

Prove that the orthocentre of the triangle whose angular points are \((a \cos \alpha, a \sin \alpha),(a \cos \beta, a \sin \beta)\) and \((a \cos \gamma, a \sin \gamma)\) is the point \([a(\cos \alpha+\cos \beta+\cos \gamma), a(\sin \alpha+\sin \beta+\sin \gamma)]\)