A ray of light incident at the point \((3,1)\) gets reflected from the tangent at \((0,1)\) to the cirde \(x^{2}+y^{2}=1\). The reflected ray touches the circle. The equation of the line along which the incident ray moves is: (a) \(3 x+4 y-13=0\) (b) \(4 x-3 y-13=0\) (c) \(3 x-4 y+13=0\) (d) \(4 x-3 y-10=0\)

\(A B C D\) is a square whose vertices \(A, B, C\) and \(D\) are \((0,0),(2,0),(2,2)\) and \((0,2)\) respectively. This square is rotated in the \(X-Y\) plane with an angle of \(30^{\circ}\) in anticlockwise direction about an axis passing through the vertex \(A\) the equation of the diagonal \(B D\) of this rotated square is......... If \(E\) is the centre of the square, the equation of the circumcircle of the triangle \(A B E\) is : (a) \(\sqrt{3} x+(1-\sqrt{3}) y=\sqrt{3}, x^{2}+y^{2}=4\) (b) \((1+\sqrt{3}) x-(1-\sqrt{2}) y=2 x^{2}+y^{2}=9\) (c) \((2-\sqrt{3}) x+y=2(\sqrt{3}-1), x^{2}+y^{2}-x \sqrt{3}-y=0\) (d) none of these

The orthocentre of the triangle with vertices $\left(2, \frac{\sqrt{3}-1}{2}\right),\left(\frac{1}{2},-\frac{1}{2}\right)$ and \(\left(2,-\frac{1}{2}\right)\) is: (a) \(\left(\frac{3}{2}, \frac{\sqrt{3}-3}{6}\right)\) (b) \(\left(2,-\frac{1}{2}\right)\) (c) \(\left(\frac{5}{4}, \frac{\sqrt{3}-2}{4}\right)\) (d) \(\left(\frac{1}{2},-\frac{1}{2}\right)\)

The cartesian co-ordinates \((x, y)\) of a point on a curve are given by $$ x: y: 1=t^{3}: t^{2}-3: t-1 $$ where \(t\) is a parameter, then the points given by \(t=a, b, c\) are collinear, if (a) \(a b c+3(a+b+c)=a b+b c+c a\) (b) \(3 a b c+2(a+b+c)=a b+b c+c a\) (c) \(a b c+2(a+b+c)=3(a b+b c+c a)\) (d) none of these

For the quadrilateral formed by \((2,-2),(8,4),(5,7)\) and \((-1,1)\) : (a) The diagonals are equal (b) the diagonals intersect at their mid points (c) the area enclosed is 36 square units (d) the radius of the circumcircle is \(3 \sqrt{\left(\frac{5}{2}\right)}\)

From the point \(A(0,3)\) on the circle \(x^{2}+4 x+(y-3)^{2}=0\), a chord \(A B\) is draw_{ } and extended to a point \(M\) such that \(A M=2 A B\). An equation of the locus of \(M\) is: (a) \(x^{2}+6 x+(y-2)^{2}=0\) (b) \(x^{2}+8 x+(y-3)^{2}=0\) (c) \(x^{2}+y^{2}+8 x-6 y+9=0\) (d) \(x^{2}+y^{2}+6 x-4 y+4=0\)