<s>[INST] Let the focus S of parabola divide one of its focal chord in the ratio 2:1. If the tangent at Q cuts directrix at R such that RQ=6... 
Let the focus S of parabola divide one of its focal chord in the ratio 2:1. If the tangent at Q cuts directrix at R such that RQ=6, find distance of focus from tangent at P.

Let P and Q be $(at^2,2at)$ and $(\frac{a}{t^2}, \frac{-2a}{t})$ respectively, for parabola $y^2=4ax$
So $$0= 2(-\frac{2a}{t})+2at$$
$$\implies t^2=2$$
Now tangent at Q is
$$\frac{y}{\sqrt 2} =x+\frac a2$$
So point R is $(-a, -\frac{a}{\sqrt 2})$
Then $$QR^2 = 36$$
$$\implies a=\frac{12}{\sqrt {11}}$$
So now, tangent at P is $$\sqrt 2y=x+2a$$ whose distance from $(a,0)$ is
$$d=\left | \frac{a+2a}{\sqrt 3}\right|$$
$$d=\sqrt 3 a$$
$$d=\frac{\sqrt 3 \times 12}{\sqrt {11}}$$
But given answer is 4
Where am I going wrong?
 [/INST] As you rightly obtained, $t^2 = 2 \implies t = \pm \sqrt2$
If we take $t = \sqrt2$ and rewrite,
$P(2a, 2a \sqrt2), Q(\frac{a}{2}, - a \sqrt2)$
Slope of tangent at point $Q = \frac{2a}{y} = - \sqrt2$
So equation of tangent line $y + a \sqrt2 = -\sqrt2(x-\frac{a}{2})$
or $y = - \sqrt2 x - \frac{a}{\sqrt2} $
Point $R$ where the line intersects the directrix, $R(-a, \frac{a}{\sqrt2})$.
So $QR^2 = \frac{9a^2}{4} + \frac{9a^2}{2} = 36 \implies a = \frac{4}{\sqrt3}$.
Now slope of tangent line at $P = \frac{2a}{y} = \frac{1}{\sqrt2}$.
So equation of tangent line at $P$, $y - 2a\sqrt2 = \frac{1}{\sqrt2}(x - 2a)$
or $y = \frac{x}{\sqrt2} + \sqrt2 a$
Its distance from focus $(a,0)$ is
$d = \frac{3a/ \sqrt2}{\sqrt{3/2}} = a \sqrt3 = 4$
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