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)`

P and Q are two points on the ellipse (x^(2))/(a^(2)) +(y^(2))/(b^(2)) =1 whose eccentric angles are differ by 90^(@) , then

P is a variable on the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 with AA' as the major axis. Find the maximum area of triangle A P A '

Let P be a point on the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 of eccentricity edot If A ,A ' are the vertices and S ,S are the foci of the ellipse, then find the ratio area P S S ' ' : area A P A^(prime)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

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 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 and Q are the foci of the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 and B is an end of the minor axis. If P B Q is an equilateral triangle, then the eccentricity of the ellipse is

If the tangent at any point of the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 makes an angle alpha with the major axis and an angle beta with the focal radius of the point of contact, then show that the eccentricity of the ellipse is given by e=cosbeta/(cosalpha)

If e_(1) is the eccentricity of the ellipse (x^(2))/(25)+(y^(2))/9=1 and if e_(2) is the eccentricity of the hyperbola 9x^(2)-16y^(2)=144 , then e_(1)e_(2) is. . . . .

Let S and S' be the foci of the ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2))=1 whose eccentricity is e. P is a variable point on the ellipse. Consider the locus the incentre of DeltaPSS' . The locus of the incenter is a\an