Line Circle

July 30, 2009 | Posted by admin

Line Circle

What is the closest distance between the line and the circle?

There is a line with a formula 7x+4y-140=0
and a circle with a formula x²+y²-8x-4y-5=0 in xy plane.
Please don’t just give the number. Also give the solution.

OK, gianlino’s answer is both correct (I checked) and simpler than mine.

Thanks to gianlino for finding my mistake.

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My thoughts on an outline for computing this:

1) Find the slope of the line (-7/4) and the center of the circle (4, 2).

2) Find the equation of a line with perpendicular slope (4/7) that passes through the center of the circle:

4/7 = (y – 2)/(x – 4)
4(x – 4) = 7(y – 2)
4x – 16 = 7y – 14
4x – 2 = 7y
y = 4/7 x – 2/7

3) Find the point where that line intersects the other, and the nearer of the two points where it intersects the circle.

Line:

4x – 7y = 2
7x + 4y = 140

16x – 28y = 8
49x + 28y = 980

65x = 988
x = 15.2

4(15.2) – 7y = 2
60.8 – 7y = 2
-7y = -58.8
y = 8.4

Circle:

x² + y² – 8x – 4y – 5 = 0
x² + (4/7x – 2/7)² – 8x – 4(4/7x – 2/7) – 5 = 0
x² + 16/49x² – 16/49x + 4/49 – 8x – 16/7x + 8/7 – 5 = 0
49x² + 16x² – 16x + 4 – 392x – 112x + 56 – 245 = 0
65x² – 520x – 185 = 0
13x² – 104x – 37 = 0

x = (104 + 14√65)/26 = (52 + 7√65)/13
y = 4/7 (52 + 7√65)/13 – 2/7 = (182 + 28√65)/91

4) Find the distance between those two points.

d = √((15.2 – (52 + 7√65)/13)² + (8.4 – (182 + 28√65)/91)²)
d = √(((145.6 – 7√65)/13)² + ((582.4 – 28√65)/91)²)
d = √((24384.36 – 2038.4√65)/169 + (390149.76 – 32614.4√65)/8281)
d = √((1194833.64 – 99881.6√65 + 390149.76 – 32614.4√65)/8281)
d = √((1584983.4 – 132496√65)/8281)
d = √(1584983.4 – 132496√65)/91
d = √516766.497/91
d = 7.8996

I tried to get it in the same form as gianlino’s, but no luck.

If you calculate both using a calculator, though, they are equal.