We are given a semi circle with radius R. We can take any point on the circumference let it be P.Now, from that point P draw two lines to the two sides of diameter. Let the lines be PQ and PS.
The task is to find the maximum value of expression PS2 + PQ for a given R.
Input : R = 1 Output : 4.25 (4*1^2 + 0.25) = 4.25 Input : R = 2 Output : 16.25 (4 * 2^2 + 0.25)= 16.25
Let F = PS^2 + PQ. We know QS = 2R (Diameter of the semicircle)
-> We also know that triangle PQS will always be right angled triangle irrespective of the position of point P on the circumference of circle
1.)QS^2 = PQ^2 + PS^2 (Pythagorean Theorem) 2.) Adding and Subtracting PQ on the RHS QS^2 = PQ^2 + PS^2 + PQ - PQ 3.) Since QS = 2R 4*R^2 + PQ - PQ^2 = PS^2 + PQ => 4*R^2 + PQ - PQ^2 = F 4.) Using the concept of maxima and minima differentiating F with respect to PQ and equating it to 0 to get the point of maxima for F i.e. 1 - 2 * PQ = 0 => PQ = 0.5 5.) Now F will be maximum at F = 4*R^2 + 0.25
- Radii of the three tangent circles of equal radius which are inscribed within a circle of given radius
- Largest rectangle that can be inscribed in a semicircle
- Largest trapezoid that can be inscribed in a semicircle
- Largest square that can be inscribed in a semicircle
- Largest triangle that can be inscribed in a semicircle
- Program to find the Area and Perimeter of a Semicircle
- Biggest Reuleaux Triangle inscirbed within a square inscribed in a semicircle
- Equation of circle from centre and radius
- Number of rectangles in a circle of radius R
- Radius of the circle when the width and height of an arc is given
- Area of a n-sided regular polygon with given Radius
- Program to find the Radius of the incircle of the triangle
- Number of common tangents between two circles if their centers and radius is given
- Find minimum radius such that atleast k point lie inside the circle
- Angular Sweep (Maximum points that can be enclosed in a circle of given radius)
If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to email@example.com. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.