Question

If the points $\left(a^{3} /(a-1),\left(a^{2}-3\right) /(a-1)\right),\left(b^{3} /(b-1)\right.$, $\left.\left(b^{2}-3\right) /(b-1)\right)$, and $\left(c^{3} /(c-1),\left(c^{2}-3\right) /(c-1)\right)$, where $a, b, c$ are different from 1 , lie on the line $l x+m y+n=0$, then

• A. $a+b+c=-\frac{m}{l}$
• B. $a b+b c+c a=\frac{n}{l}$
• C. $a b c=\frac{(m+n)}{l}$
• D. $a b c-(b c+c a+a b)+3(a+b+c)=0$

Answer 1

Answer: (A), (B), (D)

Since the given points lie on the line $l x+m y+n=0, a, b, c$ are the roots of the equation
$l\left(\frac{t^{3}}{t-1}\right)+m\left(\frac{t^{2}-3}{t-1}\right)+n=0 $
${ or } l t^{3}+m t^{2}+n t-(3 m+n)=0$
Hence, $a+b+c=-\frac{m}{l}$
$a b+b c+c a=\frac{n}{l}$
$a b c=\frac{3 m+n}{l}$
So, from (1), (2), and (3), we get
$a b c-(b c+c a+a b)+3(a+b+c)=0$