lf the eccentricity of the hyperbola `x^2 - y^2 sec^2 alpha=5` is `sqrt3` times the eccentricity of the ellipse `x^2 (sec^2alpha )+y^2=25,` then a value of `alpha` is : (a) `pi/6` (b) `pi/4` (c) `pi/3` (d) `pi/2`

The eccentricity of the ellipse 25 x^(2) + 4y^(2) = 100 is _

The eccentricity of the ellipse 4x^(2) + 25y^(2) = 100 is _

If the equation x^3+b x^2+c x+1=0,(b (a) -pi (b) -pi/2 (c) pi/2 (d) pi

If 2|sin2alpha|=|tanbeta+cotbeta|,alpha,beta in (pi/2,pi), then the value of alpha+beta is (a) (3pi)/4 (b) pi (c) (3pi)/2 (d) (5pi)/4

The eccentricity of the conjugate hyperbola of the hyperbola x^2-3y^2=1 is (a) 2 (b) 2sqrt(3) (c) 4 (d) 4/5

If the line xcostheta+ysintheta=2 is the equation of a transverse common tangent to the circles x^2+y^2=4 and x^2+y^2-6sqrt(3)x-6y+20=0 , then the value of theta is (a) (5pi)/6 (b) (2pi)/3 (c) pi/3 (d) pi/6

The distance of a point on the ellipse (x^2)/6+(y^2)/2=1 from the center is 2. Then the eccentric angle of the point is pi/4 (b) (3pi)/4 (c) (5pi)/6 (d) pi/6

The angles at which the circles (x-1)^2+y^2=10 and x^2+(y-2)^2=5 intersect is pi/6 (b) pi/4 (c) pi/3 (d) pi/2

The line tangent to the curves y^3-x^2y+5y-2x=0 and x^2-x^3y^2+5x+2y=0 at the origin intersect at an angle theta equal to (a) pi/6 (b) pi/4 (c) pi/3 (d) pi/2

let the eccentricity of the hyperbola x^2/a^2-y^2/b^2=1 be reciprocal to that of the ellipse x^2+4y^2=4. if the hyperbola passes through a focus of the ellipse then: (a) the equation of the hyperbola is x^2/3-y^2/2=1 (b) a focus of the hyperbola is (2,0) (c) the eccentricity of the hyperbola is sqrt(5/3) (d) the equation of the hyperbola is x^2-3y^2=3