Question 1097107: The line 4x+3y=25 & 3x-4y=-7 are tangent to a circle. Connecting the center of the circle, point of tangencies, and intersection of the 2 lines will form a square. The center of the circle is 6 units from the intersection of the 2 lines. Find the equation of the circle.
Answer by htmentor(1343)   (Show Source):
You can put this solution on YOUR website! First, we need to find the intersection point of the two lines
4x+3y=25 -> 16/3x + 4y = 100/3 [multiply by 4/3]
3x-4y=-7
25/3x = 100/3 - 21/3 = 79/3 = 3.16
x = 79/25
y = (25 - 4(79/25))/3 = 309/75 = 4.12
The intersection point is A = (3.16,4.12)
The length of the line segment AC, where C is the center point of the circle, = 6 units
The segment AC is a diagonal of the square connecting A, C and the two tangent points B and D
The angle between AC and CB and CD is therefore equal to 45 degrees,
since AC is a bisector of the two tangent lines meeting at A
Hence the segments CB and CD are 6/sqrt(2) in length, and this is equal to the radius of the circle.
Now we need to find the center of the circle.
Lines which are parallel to the tangent lines, and are 6/sqrt(2) units away, will go through the center of the circle.
The perpendicular distance between two parallel lines can be written as
d = |b2-b1|/sqrt(m^2+1), where m is the slope and b1, b2 the intercepts
In standard form 4x+3y=25 is y = -4/3x + 25/3, m = -4/3 b1 = 25/3
We can use this to solve for b2
d^2 = 18 = (25/3-b2)^2/((-4/3)^2+1)) = ((25/3)^2 - 50/3b + b2^2)/(25/9)
9/25b2^2 - 6b2 + 7 = 0
Solving for b2 gives 1.2623 and 15.4044, which means there are two circles which meet the conditions.
Take b2 = 1.2623. Thus one equation of a line passing through the center is
y = (-4/3)x + 1.2623 [1]
Similarly, we can find the equation of the line parallel to 3x-4y=-7, and 6/sqrt(2) units away.
These two lines will intersect at the center of the circle.
The equation of this line is y = 3/4x - 3.5533 [2]
Find the intersection point of [1] and [2]:
3/4x - 3.5533 = -4/3x + 1.2623 -> x = 2.3115
Thus y = 3/4*2.3115 - 3.5533 = -1.8197
Finally, the equation (of one) of the circle(s) is
(x-2.3115)^2 + (y+1.8197)^2 = 18
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