The line `L_1-=4x+3y-12=0` intersects the x-and y-axies at `A` and `B ,` respectively. A variable line perpendicular to `L_1` intersects the x- and the y-axis at `P` and `Q` , respectively. Then the locus of the circumcenter of triangle `A B Q` is (a) `3x-4y+2=0` (b) `4x+3y+7=0` (c) `6x-8y+7=0` (d) none of these

Let a given line L_1 intersect the X and Y axes at P and Q respectively. Let another line L_2 perpendicular to L_1 cut the X and Y-axes at Rand S, respectively. Show that the locus of the point of intersection of the line PS and QR is a circle passing through the origin

A line cuts the x-axis at A (7, 0) and the y-axis at B(0, - 5) A variable line PQ is drawn perpendicular to AB cutting the x-axis in P and the y-axis in Q. If AQ and BP intersect at R, find the locus of R

A B C is a variable triangle such that A is (1,2) and B and C lie on line y=x+lambda (where lambda is a variable). Then the locus of the orthocentre of triangle A B C is (a) (x-1)^2+y^2=4 (b) x+y=3 (c) 2x-y=0 (d) none of these

The equation of the line AB is y = x . If A and B lie on the same side of the line mirror 2x-y = 1 , then the equation of the image of AB is (a) x+y-2=0 (b) 8x+y-9=0 (c) 7x-y-6=0 (d) None of these

Two circle x^2+y^2=6 and x^2+y^2-6x+8=0 are given. Then the equation of the circle through their points of intersection and the point (1, 1) is (a) x^2+y^2-6x+4=0 (b) x^2+y^2-3x+1=0 (c) x^2+y^2-4y+2=0 (d)none of these

The center of a rectangular hyperbola lies on the line y=2xdot If one of the asymptotes is x+y+c=0 , then the other asymptote is (a) 6x+3y-4c=0 (b) 3x+6y-5c=0 (c) 3x-3y-c=0 (d) none of these

The equation of conjugate axis of the hyperbola x y-3y-4x+7=0 is (a) y+x=3 (b) y+x=7 (c) y-x=3 (d) none of these

A line is a drawn from P(4,3) to meet the lines L_1 and l_2 given by 3x+4y+5=0 and 3x+4y+15=0 at points A and B respectively. From A , a line perpendicular to L is drawn meeting the line L_2 at A_1 Similarly, from point B_1 Thus a parallelogram AA_1 B B_1 is formed. Then the equation of L so that the area of the parallelogram AA_1 B B_1 is the least is (a) x-7y+17=0 (b) 7x+y+31=0 (c) x-7y-17=0 (d) x+7y-31=0

The lines parallel to the normal to the curve x y=1 is/are (a) 3x+4y+5=0 (b) 3x-4y+5=0 (c) 4x+3y+5=0 (d) 3y-4x+5=0

A line intersects the straight lines 5x-y-4=0 and 3x-4y-4=0 at A and B , respectively. If a point P(1,5) on the line A B is such that A P: P B=2:1 (internally), find point Adot