I need math help with tangent lines and circles!?!?
The equation of the circle which is centered at (-4,-6) and which has the line -x+3y+7=0 as a tangent is
- ironduke8159Lv 71 decade agoFavorite Answer
The equation of the circle is (x+4)^2 + (y+6)^2 = r^2 where r is to be determined.
r is just the distance from (-4,-6) to the line x+3y+7= 0
This distance is given by the formula:
d = |ax1+by1+c|/sqrt(a^2+b^2), where a = 1, b= 3 and c = 7 and x1 = -4 and y1 = -6.
So d = r = |(1)((-4) + (3)(-6) + 7|/sqrt(1^2 + 3^2)
r = |-11|/sqrt(10)
So r^2 = 121/10 = 12.1
The equation is (x+4)^2 + (y+6)^2 = 12.1
- sp33dstixLv 41 decade ago
For this one you do not know the radius.. You have to find it.
let r = radius -- a constant
let (x2,y2) be the point on the line that is tangent to the circle
Now you need to find the equation of a circle:
(x+4)^2 + (y+6)^2 = r^2
You are given the tangent line:
3y = x-7
y = x/3 - 7/3
You know that a tangent line is always perpendicular to a line from the center of the circle to the point of tangent.
Given the tangent line, find a line that is perpendicular to it which contains the point (-4,-6).
Since the tangent line:
y = x/3 - 7/3 , has slope of 1/3 a perpendicular slope is -3
(y+6) = (-3)(x+4)
y = -3x -18 -- is a line that is perpendicular to the tangent and contains the mid point of the circle.
Where these two lines intersect is the point where the tangent line has a point on the circle!
Solving for x and y:
x = -47/10
y = -39/10
All that is left to do is find the distance between the midpoint and
the tangent point. This is your r value.
The rest I am sure you can figure out.
I Found r^2 = 49/10 so the equation comes out to:
(x+4)^2 + (y+6)^2 = 49/10
r^2 = 49/10
r = 7/sqrt(10) = 7*sqrt(10)/10 -- this is the most correct way of writing it.
remeber for circle:
y = +/- sqrt((49/10)-(x+4)^2) -6
I graphed out all the lines and it made sense. You can check yourself by using this program and typing in:
Type in the following (or copy paste):
Those are all the graphsSource(s): http://www.analyzemath.com/CircleEq/Tutorials.html http://www.mathwarehouse.com/geometry/circle/tange...
- fcas80Lv 71 decade ago
Distance from point (xo,yo) to line Ax + By + C = 0 is given by
| Axo + Byo + C| / SQRT(A^2 + B^2).
|1*-4 + 3*-6 +7| / SQRT(1^2 + 3^2) = 4.74
So 4.74 is radius.
Circle is (x+4)^2 + (y+6)^2 = 4.74^2