If the pair of straight lines `Ax^(2)+2Hxy+By^(2)=0` be conjugate diameters of the hyperbola `(x^(2))/(a^(2))-(y^(2))/(b^(2))=1`, then prove that `Aa^(2)=Bb^(2).`

If the lines lx+my+n=0 passes through the extremities of a pair of conjugate diameters of the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 , show that a^(2)l^(2)-b^(2)m^(2)=0 .

If the line lx+my+n=0 touches the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 . Then show a^2 l^2 - b^2 m^2 =n^2

If the pair of lines ax^2 +2hxy+ by^2+ 2gx+ 2fy +c=0 intersect on the y-axis then

Show that the tangents at the ends of conjugate diameters of the ellipse x^(2)/a^(2)+y^(2)/b^(2)=1 intersect on the ellipse x^(2)/a^(2)+y^(2)/b^(2)=2 .

If the straight line xcosalpha+ysinalpha=p touches the curve (x^2)/(a^2)-(y^2)/(b^2)=1, then prove that a^2cos^2alpha-b^2sin^2alpha=p^2dot

Prove that the locus of the middle-points of the chords of the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 which pass through a fixed point (alpha, beta) is a hyperbola whose centre is ((alpha)/(2), (beta)/(2)) .

If the pair of straight lines x^(2)-2pxy-y^(2)=0and x^(2)-2qxy-y^(2)=0 are such that each pair bisects the angle between the other pair , then prove that pq=-1 .

Length of common tangents to the hyperbolas x^2/a^2-y^2/b^2=1 and y^2/a^2-x^2/b^2=1 is