Latest Others Aptitude Question SOLUTION: Value of a for which sum of squares of roots of equation x^2-(a-2)x-a-1=0 assumes least value. A)0 B)1 C)2 D)3

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Value of a for which sum of squares of roots of equation x^2-(a-2)x-a-1=0 assumes least value.
A)0
B)1
C)2
D)3

Asked In Others harshal agrawal (10 years ago)

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Ann Theressa  (10 years ago)

a=1

A quadratic equation is of the form,
x^2 - (sum of roots)x + (product of roots) = 0

Here,
Sum of roots = a-2
product of roots = -a-1

Let the roots of the equation be p and q
p+q = a-2
p*q = -a-1

(p+q)^2 = (a-2)^2
p^2 + q^2 + 2pq = a^2 + 4 -4a
p^2 + q^2 = a^2 + 4 - 4a - 2pq

sum of squares of roots = a^2 + 4 -4a - 2(-a-1)
=a^2 +6 -2a

d/dt(a^2 + 6 -2a) = 0 [at minimum value of a]
2a - 2 = 0
a = 1

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ANUP KUMAR  (10 years ago)

a=1

let x & y are roots of the quadratic equation x^2-(a-2)x-a-1=0
then we say that, x + y = -(-(a-2))/1 = a-2
and x * y = -(a+1)/1=-(a+1);

GIVEN, x^2 + y ^2 has least value
so, x^2 + y^2=(x+y)^2-2*x*y
x^2 + y^2=(a-2)^2-2*(-(a+1))=a^2-2a+6

FOR minimum value of x^2 + y^2

d(a^2-2a+6)/d(a)= 2a-2
then d^2(2a-2)/d(a)=2 > 0
hence we can say that, at 2a-2 = 0 ,we get minmum value of a.
so, 2a-2=0
a=1

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