+1405 Find all points $(x,y)$ that are $5$ units away from the point $(2,7)$ and that lie on the line $y = 5x - 28.$
+1876 Let's note something important before we begin.
All the points 5 units away from (2, 7) forms a circle with radius 5 and center of (2, 7).
A circle with that equation would look something like
\((x-2)^2+(y-7)^2 = 5^2\)
Combining that with the second part of the problem, we have a system with
\((x-2)^2+(y-7)^2 = 25\\y=5x-28\)
Now, we already have a value for y, and we subsitute that into the first equation to isolate x.
We find that
\((x-2)^2+(5x-35)^2 = 25\)
Expanding and bringing all terms to one side of the equation, we get a quadratic. We find that
\(26x^2-354x+1204=0\)
This gives us 2 x values that will satisfy thsi equation. We get that
\(x=7,\:x=\frac{86}{13}\)
Now, we plug these two values of x back into the equation to find y, so that we can complete our problem.
We find that \(y = 7, y= \frac{66}{13}\)
So our final two abswers that satisfy all these conditions given above in the problem are
\((7, 7), (86/13, 66/13)\)
Thanks! :)
