The asymptote of the hyperbola `x^2/a^2-y^2/b^2=1` form with an tangent to the hyperbola triangle whose area is `a^2 tan lambda` in magnitude then its eccentricity is: (a) `sec lambda` (b) `cosec lambda` (c) `sec^2 lambda` (d) `cosec^2 lambda`

Portion of asymptote of hyperbola x^2/a^2-y^2/b^2 = 1 (between centre and the tangent at vertex) in the first quadrant is cut by the line y + lambda(x-a)=0 (lambda is a parameter) then (A) lambda in R (B) lambda in (0,oo) (C) lambda in (-oo,0) (D) lambda in R-{0}

From any point 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)=2. The area cut-off by the chord of contact on the asymptotes is equal to: (a) a/2 (b) a b (c) 2a b (d) 4a b

If the sum of the slopes of the normal from a point P to the hyperbola x y=c^2 is equal to lambda(lambda in R^+) , then the locus of point P is (a) x^2=lambdac^2 (b) y^2=lambdac^2 (c) x y=lambdac^2 (d) none of these

The line y=x+lambda is tangent to the ellipse 2x^(2)+3y^(2)=1 , then lambda is-

The line y=x+lambda is tangent to the ellips 2x^2+3y^2=1 Then lambda is

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.

Consider the equation (x^2)/(a^2+lambda)+(y^2)/(b^2+lambda)=1, where a and b are specified constants and lambda is an arbitrary parameter. Find a differential equation satisfied by it.

Find the locus of image of the variable point (lambda^(2), 2 lambda) in the line mirror x-y+1=0, where lambda is a parameter.

If the angle between the pair of straight lines represented by the equation x^2-3xy+lambda y^2+3x-5y+2=0 is tan^(-1) (1//3) where lambda ge 0 , then lambda is

S , T are two foci of an ellipse and B is a one end point of minor axis . If STB be a equilateral triangle and eccentricity of the ellipse be (1)/(lambda) , then the value of lambda is _