If the chord joining points `P(alpha)a n dQ(beta)` on the ellipse `((x^2)/(a^2))+((y^2)/(b^2))=1` subtends a right angle at the vertex `A(a ,0),` then prove that `tan(alpha/2)tan(beta/2)=-(b^2)/(a^2)dot`

P is any point lying on the ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2))=1(agtb) whose foci are S and S' . If anglePSS'=alpha and anglePS'S=beta , then the value of tan.(alpha)/(2)tan.(beta)/(2) is

If P(alpha,beta) is a point on the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 with foci Sa n dS ' and eccentricity e , then prove that the area of S P S ' is basqrt(a^2-alpha^2)

The tangents drawn from a point P to ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2))=1 make angles alpha and beta with the transverse axis of the hyperbola, then

If sinalpha+sinbeta and cosalpha+cosbeta=b , prove that tan(alpha-beta)/2=+-sqrt((4-a^2-b^2)/(a^2+b^2)) .

If sinalpha+sinbeta=a and cosalpha+cosbeta=b , prove that tan((alpha-beta)/2)=+-sqrt((4-a^2-b^2)/(a^2+b^2)) .

If alpha+beta=3pi , then the chord joining the points alpha and beta for the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1 passes through which of the following points? (a)Focus (b) Center (c)One of the endpoints of the transverse exis. (d)One of the endpoints of the conjugate exis.

If a chord joining P(a sec theta, a tan theta), Q(a sec alpha, a tan alpha) on the hyperbola x^(2)-y^(2) =a^(2) is the normal at P, then tan alpha =

Find the locus of the midpoint of the chords of the circle x^2+y^2=a^2 which subtend a right angle at the point (0,0)dot

If tan beta=cos theta tan alpha , then prove that tan^(2)""(theta)/(2)=(sin(alpha-beta))/(sin(alpha+beta)) .

Tangents drawn from the point (c, d) to the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 make angles alpha and beta with the x-axis. If tan alpha tan beta=1 , then find the value of c^(2)-d^(2) .