If `ax+by=1` is tangent to the hyperbola `(x^(2))/(a^(2))-(y^(2))/(b^(2))=1,` then `a^(2)-b^(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

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 point P(h, k) on the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 cuts the circle x^(2)+y^(2)=a^(2) at points Q(x_(1),y_(1)) and R(x_(2),y_(2)) , then the vlaue of (1)/(y_(1))+(1)/(y_(2)) is

If m is the slope of a tangent to the hyperbola (x^(2))/(a^(2)-b^(2))-(y^(2))/(a^(3)-b^(3)) =1 where a gt b gt 1 when

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)).

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.

The tangent at P on the hyperbola (x^(2))/(a^(2)) -(y^(2))/(b^(2))=1 meets one of the asymptote in Q. Then the locus of the mid-point of PQ is

A tangent is drawn at any point on the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2)) =1 . If this tangent is intersected by the tangents at the vertices at points P and Q, then which of the following is/are true

If y=m x+c is tangent to the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1, having eccentricity 5, then the least positive integral value of m 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 _________.