<s>[INST] Find tangent line of curve that intersects point. How do I find the tangent line of the curve $y=x^2$ that intersects the point $(8,2)$?
 [/INST] Starting with Andre's equation of
$y-a^2=2a(x-a)$,
$y=2ax-2a^2+a^2 = 2ax-a^2$,
or $a^2-2ax = -y$
or $a^2-2ax + x^2 = x^2-y$
or $(a-x)^2 = x^2-y$
or $a = x \pm \sqrt{x^2-y}$.
Note that this requires $x^2 \ge y$,
which means that the point that the tangent goes through
has to be below (or on) the parabola
and that, if the point is below, there are two tangents
going through it - see if you can visualize why.
Also note that the two values of $a$ are equally spaced around $x$.
Also$^2$ note that if $y &lt 0$
then there are always two values of $a$,
one of which is negative.
If $x=8$ and $y=2$,
$x^2-y = 62$,
so $ a = 8 \pm \sqrt{62}$.
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