Let the eccentricity of the hyperbola $${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$$ be reciprocal to that of the ellipse $${x^2} + 4{y^2} = 4$$. If the hyperbola passes through a focus of the ellipse, then

Consider the parabola $${y^2} = 8x$$. Let $${\Delta _1}$$ be the area of the triangle formed by the end points of its latus rectum and the point $$P\left( {{1 \over 2},2} \right)$$ on the parabola and $${\Delta _2}$$ be the area of the triangle formed by drawing tangents at $$P$$ and at the end points of the latus rectum. Then $${{{\Delta _1}} \over {{\Delta _2}}}$$ is

Let $$f\left( \theta \right) = \sin \left( {{{\tan }^{ - 1}}\left( {{{\sin \theta } \over {\sqrt {\cos 2\theta } }}} \right)} \right),$$ where $$ - {\pi \over 4} < \theta < {\pi \over 4}.$$

Then the value of $${d \over {d\left( {\tan \theta } \right)}}\left( {f\left( \theta \right)} \right)$$ is

The value of $$\,\int\limits_{\sqrt {\ell n2} }^{\sqrt {\ell n3} } {{{x\sin {x^2}} \over {\sin {x^2} + \sin \left( {\ell n6 - {x^2}} \right)}}\,dx} $$ is