Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
Olympiad workbookIOQMStraight Lines

Let A B C D be a unit square. Suppose M and N are points on B C and C D respectively such that the perimeter of triangle M C N is 2 . Let O be the circumcentre of triangle M A N and P be the circumcentre of triangle M O N . If ( O P O A )^2= m n for some relatively prime positive integers m and n , find the value of m+n .

Correct answer

03

Step-by-step solution

O A= circumradius of A M N= M N 2 O P= circumradius of O M N= M N 2 2 So ( O P O A )^2= ( 1 2 )^2 Perimeter of M C N=2=(1-x)+(1-y)+M N M N=x+y Now rotate A B M about A so that A B overlaps with A D (by 90^ ) Clearly A M N= A M^ N So 2 =90^ =45^ Hence ( O P O A )^2= 1 2 = m n m+n=3

Practice Straight Lines on Quantrex Academy →

More from Straight Lines

Let A B C be a triangle in the x y plane, where B is at the origin (0,0) . Let B C be produced to D such that B C: C D=1: 1, C A be produced to E such that C A: A E=1: 2 and A B beAn ant leaves the anthill for its morning exercise. It walks 4 feet east and then makes a 160^ turn to the right and walks 4 more feet. It then makes another 160^ turn to the rightA bug travels in the coordinate plane moving only along the lines that are parallel to the x axis or y axis. Let A=(-3,2) and B(3,-2) . Consider all possible paths of the bug from In parallelogram A B C D the longer side is twice the shorter side. Let X Y Z W be the quadrilateral formed by the internal bisectors of the angles of A B C D . If the area of X Y In a rectangle A B C D, E is the midpoint of A B: F is point on A C such that B F is perpendicular to A C and F E perpendicular to B D . Suppose B C=8 3 . Find A B .Consider a square A B C D of side length 16 . Let E, F be points on C D such that C E=E F=F D . Let the line BF and A E meet in M . The area of M A B is:Let A B C D be a rectangle in which A B+B C+ C D=20 and A E=9 where E is the mid-point of the side B C . Find the area of the rectangle.In the coordinate plane, a point is called a lattice point if both of its coordinates are integers. Let A be the point (12,84) . Find the number of right angled triangles A B C in Full Straight Lines list All Olympiad workbook PYQs