For the hyperbola `x^2/100 - y^2/25 = 1`. prove that (i)eccentricity = 5/2(ii) `SA.S'A = 25`, where S & S' are the foci & A is the vertex

Let 'p' be the perpendicular distance from the centre C of the hyperbola x^2/a^2-y^2/b^2=1 to the tangent drawn at a point R on the hyperbola. If S & S' are the two foci of the hyperbola, then show that (RS + RS')^2 = 4 a^2(1+b^2/p^2).

For the hyperbola 4x^(2) - 9y^(2) = 1 , find the axes, centre, eccentricity, foci and equations of the directrices.

e is the eccentricity of the ellipse 4x^(2)+9y^(2)=36 and C is the centre and S is the focus and A is the vertex then CS:SA=

A tangent is drawn at any point on the hyperbola x^(2)/a^(2) - y^(2)/b^(2) = 1 . If this tengent is intersected by the tangents at the vertices at points P and Q then show that S, S' , P and Q are concyclic points where S and S' are the foci of the hyperbola .

The eccentricity of the hyperbola x ^(2) - y^(2) =25 is

Let P be the point of intersections of the common tangents to the parabola y^(2)=12x and the hyperbola 8x^(2) -y^(2)=8. If S and S' denotes the foci of the hyperbola where S lies on the positive X-axis then P divides SS' in a ratio.

If the ellipse with equation 9x^2+25y^2=225 , then find the eccentricity and foci.