Let c and d be vectors such that | c + d |= 29 and c (2 i +3 j +4 k )=(2 i +3 j +4 k ) d . If _ 1 , _ 2 ( _ 1 > _ 2 ) are the possible values of ( c + d ) (-7 i +2 j +3 k ), then the equation K ^ 2 x^ 2 + ( K ^ 2 -5 ~K + _ 1 ) x y+ (3 ~K + _ 2 2 ) y^ 2 -8 x+12 y+ _ 2 =0 represents a circle, for K equal to :
A. 1
From c a = a d where a = 2 i + 3 j + 4 k , we get ( c + d ) a = 0. So c + d = t a . Given | c + d | = 29 : |t| 29 = 29 , so t = 1. ( c + d ) (-7 i + 2 j + 3 k ) = (-14 + 6 + 12) = 4. _1 = 4, _2 = -4. For circle: Coeff of x^2 = Coeff of y^2: K^2 = 3K - 2 K = 1, 2. Coeff of xy = 0: K^2 - 5K + 4 = 0 K = 1, 4. Common value: K = 1.
Related: Mathematics — Circle · All PYQ Banks