Three sides of a trangle have the equations $L_{r}=y-m_{r} x-c_{r}=0 ; r=1,2,3\(, Then \)\lambda L_{2} L_{3}+\mu_{3} L_{1}+v L_{1} L_{2}=0$, where \(\lambda \neq 0, \mu \neq 0, v \neq 0\), is the equation of circumcircle of triangle, if : : (a) \(\lambda\left(m_{2}+m_{3}\right)+\mu\left(m_{3}+m_{1}\right)+v\left(m_{1}+m_{2}\right)=0\) (b) $\lambda\left(m_{2} m_{3}-1\right)+\mu\left(m_{3} m_{1}-1\right)+v\left(m_{1} m_{2}-1\right)=0$ (c) both (a) and (b) (d) none of these

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

\(A(-5,0)\) and \(B(3,0)\) are two vertices of a triangle \(A B C\). Its area is $20 \mathrm{~cm}^{2}\(. The vertex \)C\( lies on the line \)x-y=2 .$ The co-ordinates of \(C\) are : (a) \((-3,-5)\) or \((-5,7)\) (b) \((-7,-5)\) or \((3,5)\) (c) \((7,5)\) or \((3,5)\) (d) \((-3,-5)\) or \((7,5)\)

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) \((-a \sqrt{3},-a \sqrt{3})\)

If the point $\left[x_{1}+t\left(x_{2}-x_{1}\right), y_{1}+t\left(y_{2}-y_{1}\right)\right]\( divides the join of \)\left(x_{1}, y_{1}\right)\( and \)\left(x_{2}, y_{2}\right)$ internally, then: (b) \(01\) (d) \(t=1\)

Length of tangent drawa from any point of the circle $x^{2}+y^{2}+2 g x+2 f y+c=0\( to the circle \)x^{2}+y^{2}+2 g x+2 f y+d=0,(d>c)$ is : (a) \(\sqrt{c-d}\) (b) \(\sqrt{d-c}\) (c) \(\sqrt{8-f}\) (d) \(\sqrt{f-8}\)