Let P : y^{2} = 4ax, a 0 be a parabola with | vedclass

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(D) Let the slope of the line $y = 3x + 5$ be $m_1 = 3$. Let the slopes of the tangents be $m$. The angle between the tangents and the line is $\frac{

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any $a > 0$

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(D) Let the slope of the line $y = 3x + 5$ be $m_1 = 3$. Let the slopes of the tangents be $m$. The angle between the tangents and the line is $\frac{\pi}{4}$.Using the formula $	an(	heta) = |\frac{m - m_1}{1 + m \cdot m_1}|$,we have $	an(\frac{\pi}{4}) = |\frac{m - 3}{1 + 3m}| = 1$.This gives two cases: $\frac{m - 3}{1 + 3m} = 1$ or $\frac{m - 3}{1 + 3m} = -1$.Case $1$: $m - 3 = 1 + 3m \implies -2m = 4 \implies m = -2$.Case $2$: $m - 3 = -1 - 3m \implies 4m = 2 \implies m = \frac{1}{2}$.Since the product of the slopes is $m_1 \cdot m_2 = (-2) \cdot (\frac{1}{2}) = -1$,the two tangents are perpendicular to each other.It is a known property of parabolas that if two tangents are perpendicular,their point of intersection lies on the directrix,and the line segment joining the points of contact $A$ and $B$ passes through the focus $S$.Therefore,$A, S,$ and $B$ are collinear for any $a > 0$.

Let $P : y^{2} = 4ax, a > 0$ be a parabola with focus $S$. Let the tangents to the parabola $P$ that make an angle of $\frac{\pi}{4}$ with the line $y = 3x + 5$ touch the parabola $P$ at $A$ and $B$. Then the value of $a$ for which $A, B$ and $S$ are collinear is:

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