If `a , b , c ,d` and `p` are distinct real numbers such that (1987, 2M) `(a^2+b^2+c^2)p^2-2(a b+b c+c d)P+(b^2+c^2+d^2)geq0,t h e na , b , c , d` are in AP (b) are in GP are in HP (d) satisfy `a b=c d`

Options

A. A.P
B. G.P
C. H.P
D. ab=cd

Explanation

To solve the problem, we need to analyze the given expression and determine the relationship between the distinct real numbers \( a, b, c, d \) and \( p \). The expression is:

\[ (a^2 + b^2 + c^2)p^2 - 2(ab + bc + cd)p + (b^2 + c^2 + d^2) \geq 0 \]

### Step 1: Recognize the Structure of the Expression
The expression resembles a quadratic in terms of \( p \). We can rewrite it as:

\[ A p^2 + B p + C \geq 0 \]

where:
- \( A = a^2 + b^2 + c^2 \)
- \( B = -2(ab + bc + cd) \)
- \( C = b^2 + c^2 + d^2 \)