Seems like this question should be straight forward, so I'm guessing it is the interpretation of the question that is messing me up.
Find the coordinates of all points on the graph of $y=1-x^2$ at which the tangent line passes thru $(2,0).$
The tangent line to $f(x)$ has a slope of $-2x$. If it passes thru $(2,0)$ then I can write the equation of the tangent line as $y=-4x+8.$ So, I just need to find the intersections of this line and the parabola function given.
Setting the 2 y values equal to find the intersections, I get,
$-4x+8=-x^2+1$ or $x^2-4x+7=0$. This quadratic has imaginary roots, so something is not right in my logic or approach or method of solution. I've checked the arithmetic. Can you assist?