For some real number , if the area of the triangle having a =3 i - j + k and b = i + j -3 k as two of its sides is 195 2 , then the number of distinct possible values of is

Step-by-step solution

a =3 i - j + k and b = i + j -3 k Area of the triangle = 195 2 1 2 | a b |= 1 2 195 | a b |^2=195 ...(i) Now, a b = | array ccc i & j & k 3 & -1 & & 1 & -3 array | aligned & = i (3- )- j (-9- ^2 )+ k (3+ ) & | a b |= (3- )^2+ (9+ ^2 )^2+(3+ )^2 aligned | a b |^2=(3- )^2+ (9+ ^2 )^2+(3+ )^2 ...(ii) From eqns. (i) and (ii), we get aligned & (3- )^2+ (9+ ^2 )^2+(3+ )^2=195 & 9+ ^2-6 +81+ ^4+18 ^2+9+ ^2+6 =195 & ^4+20 ^2+99=195 & ^4+20 ^2-96=0 & ( ^2+24 ) ( ^2-4 )=0 & ^2-4=0 or ^2+24=0 & = 2 ^2=-24 (not possible) & = 2