The equation of a circle which passes through the points of intersection of the circles \[ 2x^2 + 2y^2 - 2x + 6y - 3 = 0, \quad x^2 + y^2 + 4x + 2y + 1 = 0 \] and whose centre lies on the common chord of these circles is:
S = (-1,1) is the focus, \( 2x - 3y + 1 = 0 \) is the directrix corresponding to S and \( \frac{1}{2} \) is the eccentricity of an ellipse. If \( (a,b) \) is the centre of the ellipse, then \( 3a + 2b \) is:
If the tangent drawn at a point \( P(t) \) on the hyperbola \[ x^2 - y^2 = c^2 \] cuts the X-axis at \( T \) and the normal drawn at the same point \( P \) cuts the Y-axis at \( N \), then the equation of the locus of the midpoint of \( TN \) is:
The foot of the perpendicular drawn from \( A(1,2,2) \) onto the plane \[ x + 2y + 2z - 5 = 0 \] is \( B(a, \beta, \gamma) \). If \( \pi(x,y,z) = x + 2y + 2z + 5 = 0 \) is a plane then \(-\pi(A):\pi(B) \) is:
When \( |x| < 2 \), the coefficient of \( x^2 \) in the power series expansion of
\[ \frac{x}{(x-2)(x-3)} \]
is:
If \( y = \sin x + A \cos x \) is the general solution of \[ \frac{dy}{dx} + f(x)y = \sec x, \] then an integrating factor of the differential equation is: