Find all points $(x,y)$ that are $5$ units away from the point $(2,7)$ and that lie on the line $y = 5x - 28.$

Pythagorearn May 29, 2024

#1**+1 **

If we drew the graph for all the points 5 units away from (2, 7), we would get a circle with radius \(\sqrt5\) and center (2, 7).

A circle like that would have the equation \((x - 2)^2 + (y -7)^2 = 25\).

Now, we just have to find the intersection points with the graph of \(y = 5x - 28\).

We get a system of equations like this, and we just have to find (x, y). Luckily, y is already isolated in the second equation, so we just have to sub that into the first equation. We get

\((x -2)^2 + (5x -28 - 7)^2 = 25 \\ (x - 2)^2 + (5x - 35)^2 = 25\)

\(x^2 - 4x + 4 + 25x^2 - 350x + 1225 = 25 \\ 26x^2 - 354x + 1204 = 0\)

From this, we get\(x=86/13, 7\)

Plugging these back for y, we get \(y = 66/13, 7\).

This means our two points are \((86/13, 66/13) & (7, 7)\)

Thanks! :)

NotThatSmart May 29, 2024