`P N` is the ordinate of any point `P` on the hyperbola `(x^2)/(a^2)-(y^2)/(b^2)=1` and `A '` is its transvers axis. If `Q` divides `A P` in the ratio `a^2: b^2,` then prove that `N Q` is perpendicular to `A^(prime)Pdot`

Let P b any point on the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 whose ordinate is PN. AA′ is its transverse axis. If the point Q divides AP in the ratio a^(2):b^(2) , then NQ is?

Prove that the part of the tangent at any point of the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1 intercepted between the point of contact and the transvers axis is a harmonic mean between the lengths of the perpendiculars drawn from the foci on the normal at the same point.

Let P(4,3) be a point on the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1 . If the normal at P intersects the x-axis at (16,0), then the eccentriclty of the 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.

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 tangent 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 P Q is a double ordinate of the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1 such that O P Q is an equilateral triangle, O being the center of the hyperbola, then find the range of the eccentricity e of the hyperbola.

If two points P & Q on the hyperbola , x^2/a^2-y^2/b^2=1 whose centre is C be such that CP is perpendicularal to CQ and a lt b 1 ,then prove that 1/(CP^2)+1/(CQ^2)=1/a^2-1/b^2 .

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

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

The normal at a point P on the hyperbola b^(2)x^(2)-a^(2)y^(2)=a^(2)b^(2) of eccentricity e, intersects the coordinates axes at Q and R respectively. Prove that the locus of the mid-point of QR is a hyperbola of eccentricity (e )/(sqrt(e^(2)-1)) .