Eccentricity of ellipse \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \] if it passes through point (9, 5) and (12, 4) is:
For what value of \( k \), does the equation \[ 9x^2 + y^2 = k(x^2 - y^2 - 2x) \] represent the equation of a circle?
The principal value of \(\sin^{-1} \left( \sin \frac{5\pi}{3} \right)\) is:
A parabola has the origin as its focus and the line \( x = 2 \) as the directrix. Then the vertex of the parabola is at:
For the parabola \( y^2 = -12x \), the equation of the directrix is \( x = a \). The value of \( a \) is:
The following determinant is equal to:
The value of the integral \[ \int_0^{\frac{\pi}{2}} \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} \, dx \] is:
The equation of the hyperbola with vertices at
is: