If the equation of the hyperbola whose focus is (2, 4), eccentricity is 5 and directrix is 4x-3y+1=0 is 15x²-24xy+8y²+ax+by+c=0 then the ascending order of a, b, c is

The locus of  the point of intersection of two tangents of the hyperbola x²/a²+y²/b²=1 which  make an angle 30⁰ with one another is

A line through the origin meets the circle x²+y²=a² at P and the hyperbola x²-y²=a² at Q. The locus the point of  intersection of the tangent at P to the circle and with the tangent t Q to the hyperbola is

The equation to one asymptote of the hyperbola 14x²+38xy+20y²+x-7y-91=0 is 7x+5y-3=0, then the other asymptote is

The equation of  the transverse and conjugate axes of a hyperbola are respectively. X+2y-3=0, 2x-y+4=0 and their respective lengths are  √2 and 2/√3. The equation of the hyperbola is

Tangents to the hyperbola x²/a²+y²/b²=1 make an angles θ₁, θ₂ with the traverse axis . The equation of the locus of their intersection when tan (θ₁+θ₂)=k is

The area (in square units) of the equilateral triangle formed by the tangent at (√3, 0) to the hyperbola x²-3y²=3 with the pair of a asymptotes of the hyperbola is

The equation of the hyperbola with its transverse axis is parallel to y-axis, and its centre is (2,-3), the length of transverse axis is 12 and eccentricity 7/6 is

Tangents are drawn  from  the Point (-2, -1) to the hyperbola 2x²-3y²=6. Their equations are

If the axes are rotated through an angle of 45⁰ in the anticlockwise direction then the equation of rectangular hyperbola x²-y²=a² changes to