The area of the triangle formed by the intersection of a line parallel to X-axis and passing through $$(h, k)$$ with the lines $$y=x$$ and $$x+y=2$$ is $$4 h^{2}$$. Find the locus of point $$P$$.

Evatuate:

$$\int_\limits{0}^{\pi} e^{|\cos x|}\left[2 \sin \left(\frac{1}{2} \cos x\right)+3 \cos \left(\frac{1}{2} \cos x\right)\right] \sin x ~d x$$

Incident ray is along the unit vector $$\hat{v}$$ and the reflected ray is along the unit vector $$\widehat{w}$$. The normal is along unit vector $$\hat{a}$$ outwards. Express $$\hat{w}$$, in terms of $$\hat{a}$$ and $$\hat{v}$$.

Tangents are drawn from any point on the hyperbola $$\frac{x^{2}}{9}-\frac{y^{2}}{4}=1$$ to the circle $$x^{2}+y^{2}=9$$. Find the locus of mid-point of the chord of contact.