Q:

The $x$-coordinate of the two points $P$ and $Q$ on the parabola $y^2=8x$ are roots of $x^2-17x+11$.

The $x$-coordinates of the two points $P$ and $Q$ on the parabola $y^2=8x$ are roots of $x^2-17x+11$. If the tangents at $P$ and $Q$ meet at $T$, then find the distance of $T$ from the focus.

The point of intersection of tangents is (GM of abscissa, AM of ordinate)
Hence x coordinate of the point is 
$$x=-\sqrt{\alpha \beta}$$
$$x=-\sqrt {11}$$
And y coordinate will be 
$$y=\frac{y_1+y_2}{2}$$
But $y^2=8 \alpha$ and $y^2=8\beta$
$$y=\frac{2\sqrt {2\alpha} +2\sqrt {2\beta}}{2}$$$$y=\sqrt 2 (\sqrt{\alpha+\beta+2\sqrt {\alpha\beta}})$$
$$y=\sqrt 2 (17+2\sqrt 11)$$
As you may have already realised, I am going wrong. But I can’t seem to pinpoint it. 

A:

Note that
$$x_1+x_2=17,\>\>\>\>\>x_1x_2 =11$$
Then, the intersection coordinates of the two tangents are (GM of abscissa, AM of ordinate),
$$x=\sqrt{x_1x_2} = \sqrt{11}$$
$$y = \frac{y_1+y_2}2=\sqrt2(\sqrt{x_1}+\sqrt{x_2})=\sqrt2\sqrt{x_1+x_2+2\sqrt{x_1x_2} }
=\sqrt{34+4\sqrt{11}}$$
Since the focus of $y^2=8x$ is $(2,0)$, the distance is thus,
$$\sqrt{(x-2)^2+y^2} =\sqrt{(\sqrt{11}-2)^2+34+4\sqrt{11}}=\sqrt{49}=7$$

