=(x^(2))/(2^(2))+(y^(2))/(b^(2))=1

if two curves (x^(2))/(a^(2))+(y^(2))/(b^(2))=1&(x^(2))/(a)^(2)+(y^(2))/(b)^(2)=1 one another at right angle then

e_(1),e_(2),e_(3),e_(4) are eccentricities of the conics xy=c^(2),x^(2)-y^(2)=c^(2),(x^(2))/(a^(2))-(y^(2))/(b^(2))=1,(x^(2))/(a^(2))-(y^(2))/(b^(2))=-1 and sqrt((1)/(e_(1)^(2))+(1)/(e_(2)^(2))+(1)/(e_(3)^(2))+(1)/(e_(4)^(2)))=sec theta then 2 theta is

Prove that the curve (x^(2))/(a^(2))+(y^(2))/(b^(2))=1(x^(2))/(a^(12))+(y^(2))/(b^(2))=1 intersect orthogonally if a^(2)-a^(2)=b^(2)-b^(2)

Find the condition for the following set of curves to intersect orthogonally: (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 and xy=c^(2)(x^(2))/(a^(2))+(y^(2))/(b^(2))=1 and (x^(2))/(A^(2))-(y^(2))/(B^(2))=1

Find the condition on a and b for which two distinct chords of the hyperbola (x^(2))/(2a^(2))-(y^(2))/(2b^(2))=1 passing through (a,b) are bisected by the line x+y=b .

If the slope of a tangent to the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 is 2sqrt(2) then the eccentricity e of the hyperbola lies in the interval

PQ is a chord joining the points phi_(1) and phi_(2) on the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1. If phi_(1) and phi_(2)=2 alpha, wherealha is constant, prove that PQ touches the hyperbola (x^(2))/(a^(2))cos^(2)alpha-(y^(2))/(b^(2))=1

Show that the acute angle between the asymptotes of the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1,(a^(2)>b^(2)), is 2cos^(-1)((1)/(e)) where e is the eccentricity of the hyperbola.

If the length of minor axis of the ellipse (x^(2))/(k^(2)a^(2))+(y^(2))/(b^(2))=1 is equal to the length of transverse axis of hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 ,and the equation of ellipse is confocal with hyperbola then the value k is equal to