From the center `C` of hyperbola `(x^2)/(a^2)-(y^2)/(b^2)=1` , perpendicular `C N` is drawn on any tangent to it at the point `P(asectheta,btantheta)` in the first quadrant. Find the value of `theta` so that the area of ` C P N` is maximum.

P is a point on the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1,N is the foot of the perpendicular from P on the transverse axis. The tangent to the hyperbola at P meets the transvers axis at Tdot If O is the center of the hyperbola, then find the value of O TxO Ndot

Tangents drawn from the point (c, d) to the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 make angles alpha and beta with the x-axis. If tan alpha tan beta=1 , then find the value of c^(2)-d^(2) .

If tangents drawn from the point (a ,2) to the hyperbola (x^2)/(16)-(y^2)/9=1 are perpendicular, then the value of a^2 is _____

If the angle between the asymptotes of hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1 is 120^0 and the product of perpendiculars drawn from the foci upon its any tangent is 9, then the locus of the point of intersection of perpendicular tangents of the hyperbola can be (a) x^2+y^2=6 (b) x^2+y^2=9 (c) x^2+y^2=3 (d) x^2+y^2=18

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.

Find the area of the triangle formed by any tangent to the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1 with its asymptotes.

From the point (2, 2) tangent are drawn to the hyperbola (x^2)/(16)-(y^2)/9=1. Then the point of contact lies in the (a)first quadrant (b) second quadrant (c)third quadrant (d) fourth quadrant

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 x=asectheta,y=btantheta, then the value of x^(2)/a^(2)-y^(2)/b^(2)= …….. .

Tangent drawn from the point P(4,0) to the circle x^2+y^2=8 touches it at the point A in the first quadrant. Find the coordinates of another point B on the circle such that A B=4 .