Slope Point Form of a Line

Q. Let $P(4,-4)$ and $Q(9,6)$ be two points on the parabola $y^2=4x$ and let $X$ be any point on the arc $POQ$ of this parabola, where $O$ is the vertex of this parabola, such that the area of $△PXQ$ is maximum. Then $4$ times this maximum area (in sq. units) is

Q. Let $y=f\left(x\right)$ is a parabola of the form $y=x^2+ax+1$ and tangent to the parabola at the point of intersection with $y-$axis also touches the circle $x^2+y^2=r^2$.If it is known that no point of the parabola is below $x-$axis then the radius of circle when $a$ attains its maximum value in units is

1. $\frac{1}{\sqrt{10}}$
2. $\frac{1}{\sqrt{5}}$
3. $1$
4. $\sqrt{5}$

Q. Number of real normals to the parabola $y^2=16x,$ passing through $(4,0)$ is

1. $1$
2. $2$
3. $3$
4. $0$

Q. Through the vertex $O$ of the parabola $y^2=4ax$ a perpendicular is drawn to any tangent meeting it at $P$ and the parabola at $Q$. Then $OP\cdot OQ=$

1. $a^2$
2. $2a^2$
3. $3a^2$
4. $4a^2$