Prove that the equation of the circumcircle of the triangle formed by the lines $$ \begin{aligned} &u_{1}=a_{1} x+b_{1} y+c_{1}=0 \\ &u_{2}=a_{2} x+b_{2} y+c_{2}=0 \\ &u_{3}=a_{3} x+b_{2} y+c_{1}=0 \end{aligned} $$ $$ \text { is }\left|\begin{array}{ccc} \frac{1}{u_{1}} & \frac{1}{u_{2}} & \frac{1}{u_{3}} \\ a_{2} a_{3}-b_{2} b_{3} & a_{3} a_{1}-b_{3} b_{1} & a_{1} a_{2}-b_{1} b_{2} \\\ a_{2} b_{3}+a_{3} b_{2} & a_{3} b_{1}+a_{1} b_{3} & a_{1} b_{2}+a_{2} b_{1} \end{array}\right|=0 $$ $$ \text { or }\left|\begin{array}{ccc} \frac{a_{1}^{2}+b_{1}^{2}}{u_{1}} & \frac{a_{2}^{2}+b_{2}^{2}}{u_{2}} & \frac{a_{3}^{2}+b_{3}^{2}}{u_{3}} \\ a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \end{array}\right|=0 $$

The radical axis of the circles \(x^{2}+y^{2}+2 g x+2 f y+c=0\) and $2 x^{2}+2 y^{2}+3 x+8 y+2 c=0\( touches the circle \)x^{2}+y^{2}+2 x-2 y+1=0$. Show that either \(g=3 / 4\) or \(f=2\)

The circle \(x^{2}+y^{2}=4\) cuts the line joining the points \(A(1,0)\) and \(B(3,4)\), in two points \(P\) and \(Q\). Let \(\frac{B P}{P A}=\alpha\) and $\frac{B Q}{Q A}=\beta\( then \)\alpha\( and \)\beta$ are roots of the quadratic equation : (a) \(x^{2}+2 x+7=0\) (b) \(3 x^{2}+2 x-21=0\) (c) \(2 x^{2}+3 x-27=0\) (d) none of these

Show that the equation of a straight line meeting the circle \(x^{2}+y^{2}=a^{2}\) in two pionts at equal distances ' \(d\) ' from a point \(\left(x_{1}, y_{1}\right)\) on its circumference is $x_{1}+y y_{1}-a^{2}+\frac{d^{2}}{2}=0$

The triangle \(O A B\) is right angled where points \(O, A, B\) are $(0,0),(\cos \theta, \sin \theta)\( and \)(\cos \phi \sin \phi)\( respectively, then \)\theta$ and \(\phi\) are connected by the relation : (a) \(\sin \left(\frac{\theta-\phi}{2}\right)=\frac{1}{\sqrt{2}}\) (b) \(\cos \left(\frac{\theta-\phi}{2}\right)=\frac{1}{\sqrt{2}}\) (c) \(\sin \left(\frac{\theta-\phi}{2}\right)=-\frac{1}{\sqrt{2}}\) (d) \(\cos \left(\frac{\theta-\phi}{2}\right)=-\frac{1}{\sqrt{2}}\)

\(A B\) is a diameter of a circle. \(C D\) is a chord parallel to \(A B\) and $2 C D=A B\(. The tangent at \)B\( meets the line \)A C\( produced at \)E\(. Prove that \)A E=2 A B$.