The line y=(3x+20)/4 intersects a circle centered at the origin at A and B. We know the length of chord AB is 26. Find the area of the circle.
+208 first write \(y=\frac{3x+20}{4}\) in standard form:
\(y=\frac{3x+20}{4}\:\: \\4y=3x+20\:\:\:\:\:\:\:\:\:\: \\3x-4y+20=0\:\:\)
using distance from point to line formula:
\(\frac{\left|3\cdot 0-4\cdot 0+20\right|}{\sqrt{3^2+\left(-4\right)^2}}=\frac{20}{5}=4\)
call this distance RS
rs is a perpendicular bisector of the chord
this creates a right triangle with sides AR, RS, and AS
so the radius squared is: \(10^2+4^2=116\)
area of circle: \(\pi r^2=116\pi \)
JP
