Question

[tex] If \left|\begin{array}{ccc}a&b&ax+b\\b&c&bx+c\\ax+b&bx+c& 0\end{array}\right|

=0, then a,b,c are in.........,Select Proper option from the given options.


(a) A.P.

(b) G.P.

(c) an increasing sequence

(d) a decreasing sequence
[/tex]

Answer 1

$\left|\begin{array}{ccc}a&b&ax+b\\b&c&bx+c\\ax+b&bx+c& 0\end{array}\right|=0$

just solve it ,
=> a[c × 0 - (bx + c) × (bx + c)] - b[b × 0 -(ax +b)(bx + c) ] + (ax + b) [(bx + c) × b - (ax + b) × c] = 0

=> a{-(bx + c)²} + b(ax + b)(bx + c) + (ax +b)[b²x + bc -acx - bc ] = 0

=> -a(bx + c)² + b(ax + b)(bx + c) + (ax +b)[b²x -acx] = 0

=> (bx + c)[-a(bx + c) + b(ax + b)] + x(ax +b)(b² - ac) =0

=> (bx + c)[-abx -ac + abx + b²] + x(ax + b)(b² - ac) = 0

=> (bx + c)(b² - ac) + x(ax + b)(b² - ac) = 0

=> (b² - ac)(bx +c +ax² + bx) = 0

=> (b² - ac)(ax² + 2bx + c) = 0

now, (b² - ac) = 0 or, ax² + 2bx + c = 0
b² = ac
hence, a, b, c are in GP

so, option (b) is correct.