If the square \(A B C D\) where \(A(0,0), B(2,0), C(2,2)\) and \(D(0,2)\) undergoes the following three transformations successively (i) \(f_{1}(x, y) \rightarrow(y, x)\) (ii) \(f_{2}(x, y) \rightarrow(x+3 y, y)\) (iii) \(f_{3}(x, y) \rightarrow\left(\frac{x-y}{2}, \frac{x+y}{2}\right)\) then the final figure is a : (a) square (b) parallelogram (c) rhombus (d) none of these

The centre of a circle which is orthogonal to the circles $C_{1}: x^{2}+y^{2}=25\( and \)C_{2}:(x-7)^{2}+(y-5)^{2}=4$ and which has the least radius is : (a) \(\left(5, \frac{25}{7}\right)\) (b) \((5,7)\) (c) \((7,12)\) (d) none of these

If \(a, b, c\) are distinct real numbers, show that the points $\left(a, a^{2}\right),\left(b, b^{2}\right)\( and \)\left(c, c^{2}\right)$ are not collinear.

Find the locus of the point of intersection of two perpendicular lines each of which touches one of the two circles \((x-a)^{2}+y^{2}=b^{2},(x+a)^{2}+y^{2}=c^{2}\) and prove that the bisectors of the angles between the straight lines always touch one or the other fixed circles.

Let \(\mathrm{d}(P, O A) \leq \min \\{d(P, A B), d(P, B C), d(P, O C)\\}\) where \(d\) denotes the distance from the point to the corresponding line and \(S\) be the region consisting of all those points \(P\) inside the rectangle \(O A B C\) such that \(O=(0,0), A=(3,0), B=(3,2)\) and \(C=(0,2)\), which satisfy the above relation, then area of the region \(S\) is : (a) 2 (b) 3 \(\angle\) (c) 4 (d) none of these

\(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\) are the ends of a diameter of a circle such that \(x_{1}\) and \(x_{2}\) are the roots of $a x^{2}+b x+c=0\( and \)y_{1}\( and \)y_{2}\( are the roots of \)a_{1} y^{2}+b_{1} y+c_{1}=Q$ Find the equation of the circle and also find its centre and radius.