If the points \((2 a, a),(a, 2 a)\) and \((a, a)\) enclose a triangle of area 18 sq. units, find the centroid of the triangle.

The equation of a circle \(C_{1}\) is \(x^{2}+y^{2}=4\). The locus of the intersection of orthogonal tangents to the circle is the curve \(C_{2}\) and the locus of the intersection of perpendicular tangents to the curve \(C_{2}\) is the curve \(C_{3}\). Then : (a) \(C_{3}\) is a circle (b) The area enclosed by the curve \(C_{3}\) is \(8 \pi\) (c) \(C_{2}\) and \(C_{3}\) are circles with the same centre (d) none of these

Consider the circles and $$ \begin{aligned} &C_{1}=x^{2}+y^{2}-2 x-4 y-4=0 \\ &C_{2}=x^{2}+y^{2}+2 x+4 y+4=0 \end{aligned} $$ and the line \(L=x+2 y+2=0\). Then (a) \(L\) is the radical axis of \(C_{1}\) and \(C_{2}\) (b) \(L\) is the common tangent of \(C_{1}\) and \(C_{2}\) (c) \(L\) is the common chord of \(C_{1}\) and \(C_{2}\) (d) \(L\) is perpendicular to the line joining centres of \(C_{1}\) and \(C_{2}\)

If \(A(a, a), B(-a,-a)\) are two vertices of an equilateral triangle, then its third vertex is: (a) \(\left(\frac{a \sqrt{3}}{2},-\frac{a \sqrt{3}}{2}\right)\) (b) \((-a \sqrt{3}, a \sqrt{3})\) (c) \((a \sqrt{3},-a \sqrt{3})\) (d) \((-a \sqrt{3},-a \sqrt{3})\)

Find the equation of the pair of tangents from the origin to the circle \(x^{2}+y^{2}+2 g x+2 f y+\lambda^{2}=0\), and show that their intercept on the line \(y=h\) iss \(\frac{2 h \lambda}{\lambda^{2}-g^{2}}\) times the radius of the circle.

Find the number of integral values of \(\lambda\) for which $x^{2}+y^{2}+\lambda x+(1-\lambda) y+5=0$ is the equation of a circle whose radius can not exceed \(\underline{5}\).