Problem Solving

himu 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?

Note that the lines are parallel to each other.
Hence, the largest circle that can be inscribed such that it is tangent to both lines will have diameter equal to the shortest distance between the two lines.

Now, the most easiest method to solve this question is to use the following formula of shortest distance between two parallel lines.

• "For two non-vertical parallel lines y = mx + c1 and y = mx + c2, the shortest distance between them is given by |c2 - c1|/√(m² + 1)"

Hence, in this case, the shortest distance between the two lines = |-10 - 5|/√(2² + 1) = 15/√5 = 3√5

Hence, diameter of the circle = 3√5

Hence, area of the circle = π(3√5/2)² = (45/4)π

The correct answer is A.

Note : If the parallel lines are given as ax + by + c1 = 0 and ax + by + c2 = 0, then the shortest distance between them is given by |c2 - c1|/√(a² + b²)