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\( 28) \) 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. |
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Answer» 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 |
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