`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 locus of the center of the circle touching the line 2x-y=1 at (1,1) is (a) x+3y=2 (b) x+2y=3 (c) x+y=2 (d) none of these

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

If points Aa n dB are (1, 0) and (0, 1), respectively, and point C is on the circle x^2+y^2=1 , then the locus of the orthocentre of triangle A B C is (a) x^2+y^2=4 (b) x^2+y^2-x-y=0 (c) x^2+y^2-2x-2y+1=0 (d) x^2+y^2+2x-2y+1=0

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

Parabola y^2=4a(x-c_1) and x^2=4a(y-c_2) , where c_1a n dc_2 are variable, are such that they touch each other. The locus of their point of contact is (a) x y=2a^2 (b) x y=4a^2 (c) x y=a^2 (d) none of these

A triangle A B C with vertices A(-1,0),B(-2,3/4), and C(-3,-7/6) has its orthocentre at Hdot Then, the orthocentre of triangle B C H will be (-3,-2) (b) 1,3) (-1,2) (d) none of these

The circle x^2+y^2-4x-4y+4=0 is inscribed in a variable triangle O A Bdot Sides O A and O B lie along the x- and y-axis, respectively, where O is the origin. Find the locus of the midpoint of side A Bdot

P is a point on the line y+2x=1, and Q and R two points on the line 3y+6x=6 such that triangle P Q R is an equilateral triangle. The length of the side of the triangle is (a) 2/(sqrt(5)) (b) 3/(sqrt(5)) (c) 4/(sqrt(5)) (d) none of these

In triangle ABC, the coordinates of the vertex A are , (4,-1) and lines x - y - 1 = 0 and 2x - y = 3 are the internal bisectors of angles B and C . Then the radius of the circles of triangle AbC is

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