Answer
$\boxed{(2, 4), (4, 16)}$
Work Step by Step
Given the function $f(x) = x^2$, we need to find all points $(x, y)$ on the graph of $f(x)$ with tangent lines passing through the point $(3, 8)$.
Step 1: Find the first derivative of the function.
$f'(x) = \frac{d}{dx}(x^2) = 2x$
Step 2: Find the equation of the tangent line passing through the point $(x_0, y_0)$ on the graph of $f(x)$.
The slope of the tangent line is $f'(x_0) = 2x_0$.
The equation of the tangent line is:
$y - y_0 = 2x_0(x - x_0)$
$y = 2x_0(x - x_0) + y_0$
Step 3: Find the values of $x_0$ such that the tangent line passes through the point $(3, 8)$.
Substituting $(3, 8)$ into the equation of the tangent line, we get:
$8 = 2x_0(3 - x_0) + x_0^2$
Solving this quadratic equation, we get:
$x_0 = 2, 4$
Step 4: Evaluate the function at the corresponding $x_0$ values to find the points $(x, y)$.
For $x_0 = 2$:
$y_0 = f(2) = 2^2 = 4$
The point is $(2, 4)$.
For $x_0 = 4$:
$y_0 = f(4) = 4^2 = 16$
The point is $(4, 16)$.
Therefore, the points $(x, y)$ on the graph of $f(x) = x^2$ with tangent lines passing through the point $(3, 8)$ are:
$\boxed{(2, 4), (4, 16)}$
