[email protected] wrote:Given the two lines y = 2x + 5 and y = 2x - 10, what is the area of the largest circle that can be inscribed such that it is tangent to both lines!
Answer is 45pi
y = 2x + 5 ____________(1)
y = 2x - 10 ____________(2)
Both the lines are of the form y = mx + c.
Clearly, slopes m are the same.
Since the slopes are the same the two lines are parallel.
This means the greatest diameter the circle can have with the conditions given is equal to the distance between the 2 lines.
My solution:
To find the distance, I used a line perpendicular to both these lines: y = (-1/2)x ____________(3)
Solving (1) and (3):
(-1/2)x = 2x + 5
(-5/2)x = 5
x = -2
y = 1
Solving (2) and (3):
(-1/2)x = 2x - 10
(5/2)x = 10
x = 4
y = -2
So diameter = distance between (-2,1) and (4,-2) = sqrt(((-2)-4)^2 + (1-(-2))^2)
= sqrt(36 + 9)
= sqrt(45)
Radius R = sqrt(45)/2
Area = pi * R^2
= 45pi/4
