Question

Two circles of radius 6 are tangent to the graph of y^2 = 4x at the point (1, 2). Find equations of these two circles. (Enter your answers as a comma-separated list.)

Answer 1

Step 1

If both circles are tangent to the function y²=4x at(1,2) then they must have the same slope at (1,2) that y²=4x has at (1,2). So we will find the slope of the tangent line to y²=4x at (1,2).

$\begin{gathered} y^2=4x \\ \text{differentiating explictly we have;} \\ \frac{d(y^2)}{dx}=\frac{d(4x)}{dx} \\ 2y(\frac{dy}{dx})=4 \\ \frac{dy}{dx}=\frac{4}{2y} \\ \frac{dy}{dx}=\frac{2}{y} \end{gathered}$$\begin{gathered} \text{At }(1,2)\text{ } \\ \frac{dy}{dx}=\frac{2}{2}=1 \end{gathered}$

Step 2

Let us assume the upper circle has a center at (x₁,y₁) r