Length of intercept cut off from a line by a Circle
Given six integers, a, b, c, i, j, and k representing the equation of the circle and equation of the line , the task is to find the length of the intercept cut off from the given line to the circle.
Input: a = 0, b = 0, c = -4, i = 2, j = -1, k = 1
Input: a = 5, b = 6, c = -16, i = 1, j = 4, k = 3
Approach: The problem can be solved based on the following mathematical fact.
As the line is intercepted by the circle the intercepted segment forms a chord of the circle. Now if a perpendicular is drawn from the centre of the circle to a chord, it divides the chord into two equal parts.
We can easily calculate the radius of the circle and the perpendicular distance of the chord from the centre from the given equations Using them we can get the length of the intercepted line segment.
Follow the steps below to solve the problem:
- Find the center of the circle, say as and .
- The perpendicular from the center divides the intercept into two equal parts, therefore calculate the length of one of the parts and multiply it by 2 to get the total length of the intercept.
- Calculate the value of radius (r) using the formula: , where and
- Calculate the value of perpendicular distance ( d ) of center O from the line by using the formula:
- Now from the Pythagoras theorem in triangle OCA:
- After completing the above steps, print the value of twice of AC to get the length of the total intercept.
Below is the implementation of the above approach.
Time Complexity: O(logN), for using in-built sqrt() function.
Auxiliary Space: O(1)