Paragraph: Consider A B C whose vertices are A (m, n), B (1,2), C (2,3) and vertex 'A' lies on the line 2 x-y+3=0 where m, n N with m^2+n^2=90 . Let area of A B C be S such that [S]=2 , where [x] denotes greatest integer less than or equal to x . Question: If the point R( , ) lies inside the A B C is such that the A B R, B C R and C A R are of equal area, then ( 2 +3 ) equals

Step-by-step solution

aligned & As A(m, n) lies on y=2 x+3 & n=2 m+3 & A=(m, 2 m+3) aligned Area of ABC = 1 2 | array ccc m & 2 ~m +3 & 1 1 & 2 & 1 2 & 3 & 1 array |= 1 2 | ~m +2|= S (Given) aligned & But [ S ]=2 2 S 3 & i.e., 2 1 2 | ~m +2| 3 aligned 4 |m+2| 6 Now, | m +2| 6 -6 m +2 6 -8 m 4 (1) and |m+2| 4 m+2 4 or m+2 -4 m 2 or m -6 From (i) and (ii), we get -8 m -6 or 2 m 4 m =-7,-6,2,3 But m N , So, m -7,-6 aligned & m=2,3 & Also m=2 n=2 m+3=2(2)+3=7 & A(2,7) aligned aligned & But m ^2+ n ^2=90 and 2^2+7^2 90 & ~m 2 & Now m =3 n =2