For p,q 
R, consider the real valued function f(x) = (x - p) 2 -q, x 
R and q > 0. Let a 1 , a 2 , a 3 and a 4 be in an arithmetic progression with mean p and positive common difference. If |f(a i )| = 500 for all i = 1, 2, 3, 4, then the absolute difference between the roots of f(x) = 0 is
Text Solution
Verified by Experts50
(50)
f(x) = 0 
(x - p) 2 - q = 0.
Roots are p + 
, p - 
absolute difference
between roots 2 
.
Now, |f(a i )| =500
Let a 1 , a 2 , a 3 , a 4 are a 1 a + d, a + 2d, a + 3d
|f(a 4 )| = 500
|(a 1 - p) 2 - q| = 500
d 2 - q = 500 __________(1)
and |f(a 1 )| 2 = |f(a 2 )| 2
((a 1 - p) 2 - q) 2 = ((a 2 - p) 2 - q) 2
((a 1 -p) 2 -(a 2 -p) 2 )((a 1 -p) 2 -q + (a 2 -p) 2 -q) = 0
From equation (1) 
= 500
──────────────────────────────────────────────────────────────────────────────────────────
Prepare Smarter with CGP Edu
Get practice questions, solutions, and test series in one place.
Write a Review
Share your experience with this question and solution.
Commentary
Send your comment, doubt, correction, or feedback to admin.
Similar Questions
Explore conceptually related problems