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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.

Jul 14, 2021

#1
+208
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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

Jul 14, 2021
#2
+124525
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Just  a slight  mistake  by JP....

We  have   a right triangle   with   legs of  4  and 13

The  radius^2   =   4^2  + 13^2   =    185

So.....the  area  =   185 pi

Jul 14, 2021
#3
+208
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thanks for the correction

JKP1234567890  Jul 14, 2021