If the normal at `P(theta)` on the hyperbola `(x^2)/(a^2)-(y^2)/(2a^2)=1` meets the transvers axis at `G ,` then prove that `A GdotA^(prime)G=a^2(e^4sec^2theta-1)` , where `Aa n dA '` are the vertices of the hyperbola.

If the normal at any point P on the ellipse x^2/a^2+y^2/b^2=1 meets the axes at G and g respectively, then find the ratio PG:Pg .

A normal to the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1 meets the axes at Ma n dN and lines M P and N P are drawn perpendicular to the axes meeting at Pdot Prove that the locus of P is the hyperbola a^2x^2-b^2y^2=(a^2+b^2)^2 dot

The tangent at a point P on the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1 passes through the point (0,-b) and the normal at P passes through the point (2asqrt(2),0) . Then the eccentricity of the hyperbola is

If e is the eccentricity of the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 and theta is the angle between the asymptotes, then cos.(theta)/(2) is equal to

If the tangent at the point (2sec theta,3tan theta) to the hyperbola (x^(2))/(4)-(y^(2))/(9)=1 is parallel to 3x-4y+4=0 , then the value of theta , is

P is a point on the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 , and N is the foot of the perpendicular from P on the transverse axis. The tantent to the hyperbola at P meets the transverse axis at T. If O is the centre of the hyperbola, then OT.ON is equal to

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).

Find the equations of the tangent and normal to the hyperbola x^2/a^2 - y^2/b^2 = 1 at the point (x_0,y_0)