If the line $y = mx + c$ is a tangent to the parabola $y^2 = 4a(x + a)$,then $ma + \frac{a}{m}$ is equal to

If the normal chord drawn at the point $\left(\frac{15}{2}, \frac{15}{\sqrt{2}}\right)$ to the parabola $y^2=15x$ subtends an angle $\theta$ at the vertex of the parabola,then $\sin \frac{\theta}{3}+\cos \frac{2\theta}{3}-\sec \frac{4\theta}{3}=$

The equation of the given curve is $x^2-4x+4y-8=0$. Match the following:

List-$I$	List-$II$
$(A)$ Focus	$(I)$ $(4,2)$
$(B)$ Vertex	$(II)$ $(3,2)$
$(C)$ One end of the latus rectum	$(III)$ $(2,3)$
$(D)$ Point of intersection of the axis and directrix	$(IV)$ $(2,4)$
	$(V)$ $(2,2)$

The correct matching is:

Find the locus of the midpoints of the chords of the parabola $x^2 + 4y = 0$ which pass through its focus.

An equilateral triangle is inscribed in the parabola $y^{2}=4ax$,where one vertex is at the vertex of the parabola. Find the length of the side of the triangle.

The locus of the feet of the perpendiculars drawn from the vertex of the parabola $y^2 = 4ax$ upon all such chords of the parabola which subtend a right angle at the vertex is