+1557 Find all points $(x,y)$ that are $5$ units away from the point $(2,7)$ and that lie on the line $y = 5x - 28.$
+868 Let the point $(x, y)$ be $5$ units away from the point $(2, 7)$. This means that the distance between $(x, y)$ and $(2, 7)$ is $5$ units.
Using the distance formula, we have
\[\sqrt{(x - 2)^2 + (y - 7)^2} = 5.\]
Squaring both sides, we get
\[(x - 2)^2 + (y - 7)^2 = 25.\]
Since the point $(x, y)$ lies on the line $y = 5x - 28$, we substitute $y = 5x - 28$ into the equation above and solve for $x$:
\[(x - 2)^2 + (5x - 35)^2 = 25.\]
\[x^2 - 4x + 4 + 25x^2 - 350x + 1225 = 25.\]
\[26x^2 - 354x + 1205 = 0.\]
The solutions to this quadratic equation for $x$ are $x = 5$ and $x = \frac{23}{13}$. Plugging these values of $x$ back into the equation $y = 5x - 28$, we find that the corresponding $y$ values are $y = 2$ and $y = 1$, respectively.
Therefore, the points that are $5$ units away from $(2, 7)$ and lie on the line $y = 5x - 28$ are $(5, 2)$ and $\left(\frac{23}{13}, 1\right)$.
