(x-a)(x-c) + 2(x-b)(x-d) = 0
or, 3x2 - (a+c+2d+2b)x + ac +2bd =0
from, this find the discriminant..
see what are the conditions u get for, b2-4ac≥0
would be a lenghty one.
For real x, the function (x-a)(x-b)/(x-c) will assume all real values provided
a) a>b>c
b) a<b<c
c) a>c>b
d) a<c<b
True / False (with explanation)
If a<b<c<d, then the roots of the equation,
(x-a)(x-c) + 2(x-b)(x-d) = 0 are real and distinct
let (x-a)(x-b)(x-c) = y
then x2 - (a+b) x + ab = xy - cy
thus, x2 - (a+b+y) x + ab +cy = 0 .....1
as x is real, then, discriminant ≥ 0
then,
(a+b+y)2 - 4ab - 4cy ≥ 0
a2+b2+2ab + y2+ 2ay + 2by -4ab - 4cy≥0
y2 + 2(a+b-2c) y +(a-b)2 ≥ 0 ....2
as expression 2 is positive,
discriminant should be negetive,
thus,
4(a+b-2c)2 - 4(a-b)2 ≤ 0
or, (a+b-2c+a-b) (a+b-2c-a+b) ≤ 0
0r, (a-c)(b-c)≤0
thus, c lies between a and b.
thus, (c) and (d) should be tha answers.. :)
(x-a)(x-c) + 2(x-b)(x-d) = 0
or, 3x2 - (a+c+2d+2b)x + ac +2bd =0
from, this find the discriminant..
see what are the conditions u get for, b2-4ac≥0
would be a lenghty one.
true/false
The points with position vectors a+b, a-b,and a+kb are collinear for all values of k.
1984-2
The correct way of doing this would be--
consider f(x) = (x-a)(x-c)+2(x-b)(x-d)
f(t1) >0 ; t1>d
f(c) <0
f(t2) >0 ; t2<a
By Intermediate Value theorem there is one root of f(x)=0 in the interval (t1,c)
and another root in the interval (c,t2)
so f(x) = 0 has real and distinct roots.