91Ó°ÊÓ

Find the equation of the circle circumscribing the triangle formed by the axes and the straight line \(3 x+4 y+12=0\).

If one end of the diameter of the circle \(x^{2}+y^{2}-2 x+6 y-15=0\) is \((4,1)\), find the coordinates of the other end.

Find the slope of the radius of the circle \(x^{2}+y^{2}=25\) through the point \((3,-4)\) and hence write down the equation of the tangent to the circle at the point. What are the intercepts made by this tangent on the \(x\) -axis and \(y\) -axis?

Prove that the centres of the three circles \(x^{2}+y^{2}-2 x+6 y+1=0\), \(x^{2}+y^{2}+4 x-12 y-9=0\) and \(x^{2}+y^{2}=25\) lie on the same straight line. What is the equation of this line?

Find the equation of the locus of a point that moves in a plane so that the sum of the squares from the line \(7 x-4 y-10=0\) and \(4 x+7 y+5=0\) is always equal to 3 .

Find the equation for the circle concentric with the circle \(x^{2}+y^{2}-8 x+6 y-5=0\) and passes through the point \((-2,7) .\)

If the distances of origin to the centres of three circles \(x^{2}+y^{2}-2 \lambda x=c^{2}\) where \(\lambda\) is a variable and \(c\) is a constant are in G. P, prove that the length of the tangent drawn to them from any point on the circle \(x^{2}+y^{2}=c^{2}\) are in G. P.

If the pole of any line with respect to the circle \(x^{2}+y^{2}=a^{2}\) lies on the circle \(x^{2}+y^{2}=9 a^{2}\), then show that the line will be a tangent to the circle \(x^{2}+y^{2}=\frac{a^{2}}{9}\)

Find the equation of the circle that touches the \(y\) -axis at a distance of 4 units from the origin and cuts off an intercept of 6 units from the \(x\) -axis.

A point moves so that the sum of the squares of the perpendiculars that fall from it on the sides of an equilateral triangle is constant. Prove that the locus is a circle.