Normal are drawn to the hyperbola `(x^2)/(a^2)-(y^2)/(b^2)=1` at point `theta_1a n dtheta_2` meeting the conjugate axis at `G_1a n dG_2,` respectively. If `theta_1+theta_2=pi/2,` prove that `C G_1dotC G_2=(a^2e^4)/(e^2-1)` , where `C` is the center of the hyperbola and `e` is the eccentricity.

C is the center of the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1 The tangent at any point P on this hyperbola meet the straight lines b x-a y=0 and b x+a y=0 at points Qa n dR , respectively. Then prove that C QdotC R=a^2+b^2dot

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 a x+b y=1 is tangent to the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1 , then a^2-b^2 is equal to 1/(a^2e^2) (b) a^2e^2 b^2e^2 (d) none of these

Locus of perpendicular from center upon normal to the hyperbola (x^(2))/(a^(2)) -(y^(2))/(b^(2)) =1 is

If the angle between the asymptotes of hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 id (pi)/(3) , then the eccentnricity of conjugate hyperbola is _________.

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 any line perpendicular to the transverse axis cuts the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1 and the conjugate hyperbola (x^2)/(a^2)-(y^2)/(b^2)=-1 at points Pa n dQ , respectively, then prove that normal at Pa n dQ meet on the x-axis.

If the normal at a pont P to the hyperbola x^2/a^2 - y^2/b^2 =1 meets the x-axis at G , show that the SG = eSP.S being the focus of the hyperbola.

Let P(6,3) be a point on the hyperbola parabola x^2/a^2-y^2/b^2=1 If the normal at the point intersects the x-axis at (9,0), then the eccentricity of the hyperbola is

Statement 1 : If from any point P(x_1, y_1) on the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=-1 , tangents are drawn to the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1, then the corresponding chord of contact lies on an other branch of the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=-1 Statement 2 : From any point outside the hyperbola, two tangents can be drawn to the hyperbola.