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`

The eccentricity of the hyperbola x^2-y^2=4 is

Prove that the locus of the point of intersection of the tangents at the ends of the normal chords of the hyperbola x^2-y^2=a^2 is a^2(y^2-x^2)=4x^2y^2dot

Prove that the locus of the point of intersection of the tangents at the ends of the normal chords of the hyperbola x^(2)-y^(2)=a^(2)" is " a^(2)(y^(2)-x^(2))=4x^(2)y^(2) .

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

Find the common tangents to the hyperbola x^(2)-2y^(2)=4 and the circle x^(2)+y^(2)=1

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 equation of a tangent to the hyperbola 3x^(2)-y^(2)=3 , parallel to the line y = 2x +4 is

Show that the locus of the midpoints of the chords of the circle x^(2)+y^(2)=a^(2) which are tangent to the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 is (x^(2)+y^(2))^(2)=a^(2)x^(2)-b^(2)y^(2) .

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

Find the equation of tangents to hyperbola x^(2)-y^(2)-4x-2y=0 having slope 2.