Question

\( 28) \) If \( \sin \alpha, \cos \alpha \) are the roots of the equation \( a x^{2}+b x+c-0(c \neq 0) \), then prove that \( (n+c)^{2}-b^{2}+c^{2} \) 29) Find value of a for which the sum of the squares of the equation \( x^{2}-(a-2) x-a-1=0 \) assumes the least value.

Answer 1

28} Let the roots of the equation ax2+bx+c= 0(where c≠0),  be p and q

So by the given condition  p = sinα and q = cosα

and p+q = -b/a and pq = c/a

again p² +q² = (p+q)2 -2pq

=> sin²α + cos²α = (-b/a)2 - 2c/a

=> 1= (b²-2ca)/a²

=>a²=(b²-2ca)

=>a² +2ca +c2=b² +c2

=>(a +c)²=b² +c2 proved

29) Let the roots of the quadratic equation x²−(a−2)x−(a+1)=0 be p and q

then p+q =a-2 and pq = - (a+1)

So p²+q² = (p+q)2-2pq= ( a-2)²+2(a+1)=a²-4a +4+2a+2=a²-2a+6 = (a-1)²+5

It is obvious that the value of p²+q² will be least only when a-1 = 0 or a =1