The ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$ and the straight line $y = mx + c$ intersect in real points only if
${a^2}{m^2} < {c^2} - {b^2}$
${a^2}{m^2} > {c^2} - {b^2}$
${a^2}{m^2} \ge {c^2} - {b^2}$
$c \ge b$
If the variable line $y = kx + 2h$ is tangent to an ellipse $2x^2 + 3y^2 = 6$ , the locus of $P (h, k)$ is a conic $C$ whose eccentricity equals
The lengths of major and minor axis of an ellipse are $10$ and $8$ respectively and its major axis along the $y$ - axis. The equation of the ellipse referred to its centre as origin is
The equation of an ellipse whose eccentricity is $1/2$ and the vertices are $(4, 0)$ and $(10, 0)$ is
The normal at $\left( {2,\frac{3}{2}} \right)$ to the ellipse, $\frac{{{x^2}}}{{16}} + \frac{{{y^2}}}{3} = 1$ touches a parabola, whose equation is
The tangent and normal to the ellipse $3x^2 + 5y^2 = 32$ at the point $P(2, 2)$ meet the $x-$ axis at $Q$ and $R,$ respectively. Then the area(in sq. units) of the triangle $PQR$ is