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

magenta Aug 9, 2024

#1**+1 **

First, let's make some observations about this question.

All points 5 units away from point (2, 7) would make a circle with center (2, 7) and radius 5.

The formula for that circle would look something like \((x-2)^2+(y-7)^2=25\)

Now, combining that with the second equation, and we get a system.

Subsituting out y with the x values in the line equation, we get

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

Simplifying, and we get

\(13x^{2}-177x+602=0\)

Using the quadratic equation, we find two values of x. We get

\(x=7\\ x=\frac{86}{13}\)

Plugging these x values into the second equation where we isolated y, we get that \(y=7,y=\frac{66}{13}\)

So our final two answers are (7, 7) and (86/13, 66/13)

Thanks! :)

NotThatSmart Aug 9, 2024