Can someone show me how to find the points that intersect with the graph of the function f(x) = 1-x^2 and the tangent line that goes through the point (2,0).
The point is on the curve $\displaystyle f(x)=1-x^2 $. Say the point is $\displaystyle (a,1-a^2)$. The slope of the tangent line is $\displaystyle k=f'(a)=-2a$. So the tangent line is
$\displaystyle f(x)=-2a(x-a)+1-a^2=-2ax+1+a^2$
The line passes through the point $\displaystyle (2,0) $, then $\displaystyle 0=-4a+1+a^2$
Therefore, $\displaystyle a= 2\pm\sqrt{3}$