Let p be the number of all triangles that can be formed by joining the vertices of a regular polygon P of n sides and q be the number of all quadrilaterals that can be formed by joining the vertices of P. If p + q = 126, then the eccentricity of the ellipse x²/16 + y²/n = 1 is:
Question Let p p be the number of all triangles that can be formed by joining the vertices of a regular polygon P P of n n sides, and q q be the number of all quadrilaterals that can be formed by joining the vertices of P P . If p + q = 126 , then find the eccentricity of the ellipse x 2 16 + y 2 n = 1. Question Image Short Solution Triangles from n n n vertices: p = ( n 3 ) Quadrilaterals from n n n vertices: q = ( n 4 ) Given: ( n 3 ) + ( n 4 ) = 126 Solve for n n . Substitute n n n as the denominator of y 2 y^{2} in the ellipse: x 2 16 + y 2 n = 1 Identify a a and b b , then find eccentricity: e = 1 − b 2 a 2 Image Solution Conclusion Step 1: Solve for n n n ( n 3 ) = n ( n − 1 ) ( n − 2 ) 6 , ( n 4 ) = n ( n − 1 ) ( n − 2 ) ( n − 3 ) 24 So: n ( n − 1 ) ( n − 2 ) 6 + n ( n − 1 ) ( n − 2 ) ( n − 3 ) 24 = 126 Multiply by 24: 4 n ( n − 1 ) ( n − 2 ) + n ( n − 1 ) ( n − 2 ) ( n − 3 ) = 3024 Factor n ( ...