If p and q are the lengths of the perpendiculars from the origin on the lines,

x cosec $$\alpha$$ $$-$$ y sec $$\alpha$$ = k cot 2$$\alpha$$ and

x sin$$\alpha$$ + y cos$$\alpha$$ = k sin2$$\alpha$$

respectively, then k² is equal to :

Let A be a fixed point (0, 6) and B be a moving point (2t, 0). Let M be the mid-point of AB and the perpendicular bisector of AB meets the y-axis at C. The locus of the mid-point P of MC is :

Let ABC be a triangle with A($$-$$3, 1) and $$\angle$$ACB = $$\theta$$, 0 < $$\theta$$ < $${\pi \over 2}$$. If the equation of the median through B is 2x + y $$-$$ 3 = 0 and the equation of angle bisector of C is 7x $$-$$ 4y $$-$$ 1 = 0, then tan$$\theta$$ is equal to :

The point P (a, b) undergoes the following three transformations successively :

(a) reflection about the line y = x.

(b) translation through 2 units along the positive direction of x-axis.

(c) rotation through angle $${\pi \over 4}$$ about the origin in the anti-clockwise direction.

If the co-ordinates of the final position of the point P are $$\left( { - {1 \over {\sqrt 2 }},{7 \over {\sqrt 2 }}} \right)$$, then the value of 2a + b is equal to :