A triangle is formed by the lines `x+y=0,x-y=0,` and `l x+m y=1.` If `la n dm` vary subject to the condition `l^2+m^2=1,` then the locus of its circumcenter is (a) `(x^2-y^2)^2=x^2+y^2` (b) `(x^2+y^2)^2=(x^2-y^2)` (c) `(x^2+y^2)^2=4x^2y^2` (d) `(x^2-y^2)^2=(x^2+y^2)^2`

The locus of a point, from where the tangents to the rectangular hyperbola x^2-y^2=a^2 contain an angle of 45^0 , is (a) (x^2+y^2)^2+a^2(x^2-y^2)=4a^2 (b) 2(x^2+y^2)^2+4a^2(x^2-y^(2))=4a^2 (c) (x^2+y^2)^2+4a^2(x^2-y^2)=4a^2 (d) (x^2+y^2)+a^2(x^2-y^(2))=a^4

Show that (x^2+y^2)^4=(x^4-6x^2y^2+y^4)^2+(4x^3y-4x y^3)^2dot

If points A and B 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

A circle of constant radius a passes through the origin O and cuts the axes of coordinates at points P and Q . Then the equation of the locus of the foot of perpendicular from O to P Q is (A) (x^2+y^2)(1/(x^2)+1/(y^2))=4a^2 (B) (x^2+y^2)^2(1/(x^2)+1/(y^2))=a^2 (C) (x^2+y^2)^2(1/(x^2)+1/(y^2))=4a^2 (D) (x^2+y^2)(1/(x^2)+1/(y^2))=a^2

If cos^(-1)x+cos^(-1)y+cos^(-1)z=pi,t h e n (a) x^2+y^2+z^2+x y z=0 (b) x^2+y^2+z^2+2x y z=0 (c) x^2+y^2+z^2+x y z=1 (d) x^2+y^2+z^2+2x y z=1

If sin^(-1)((x^2-y^2)/(x^2+y^2))=loga ,t h e n(dy)/(dx) is equal to (a) x/y (b) y/(x^2) (C) (x^2-y^2)/(x^2+y^2) (d) y/x

From the points (3, 4), chords are drawn to the circle x^2+y^2-4x=0 . The locus of the midpoints of the chords is (a) x^2+y^2-5x-4y+6=0 (b) x^2+y^2+5x-4y+6=0 (c) x^2+y^2-5x+4y+6=0 (d) x^2+y^2-5x-4y-6=0

The locus of a point reprersented by x=a/2((t+1)/t),y=a/2((t-1)/t) , where t in R-{0}, is (a) x^2+y^2=a^2 (b) x^2-y^2=a^2 (c) x+y=a (d) x-y=a

The equation of the circle which has normals (x-1).(y-2)=0 and a tangent 3x+4y=6 is (a) x^2+y^2-2x-4y+4=0 (b) x^2+y^2-2x-4y+5=0 (c) x^2+y^2=5 (d) (x-3)^2+(y-4)^2=5

Find the radical center of the circles x^2+y^2+4x+6y=19 ,x^2+y^2=9,x^2+y^2-2x-4y=5,