How do I solve for a point on a curve when I am given the curve, and the information the tangent line runs through a point $(x, y)$?

Question

For example, $y = \frac{1}{x^2 + 1}$ and the tangent runs through the point $(x, y)$. Find a point on the curve $y$.

I know I should take the derivative and find the tangent. However, I am unsure of how to do so?

Answer 1

$x,y$ is a point on the curve.

$y = \frac {1}{1+x^2}$

The slope of the tangent this point

$y'= \frac {-2x}{(1+x^2)^2}$

and the tangent runs through the point $x_0,y_0$

then $\frac {y - y_0}{x - x_0} = y'$

$\frac {\frac {1}{1+x^2} - y_0}{x - x_0} = \frac {-2x}{(1+x^2)^2}$

and solve for $x$

Answer 2

For a given $x$ you have a point on the curve $(x,\frac 1{1+x^2})$ and the slope at that point $y'(x)$. There is a point-slope equation for a line which you can use to write the equation of the tangent line for any given $x$. Now plug in the point the tangent is to run through. You will get an equation in $x$, the $x$ coordinate on the curve that the tangent touches at. Solve it.