If the points (a3 /(a-1),(a2-3) /(a-1)),(b3 /(b-1)., .(b2-3) /(b-1)), and (c3 /(c-1),(c2-3) /(c-1)), where a, b, c are different from 1 , lie on the line l x+m y+n=0, then

Question

If the points (a3 /(a-1),(a2-3) /(a-1)),(b3 /(b-1)., .(b2-3) /(b-1)), and (c3 /(c-1),(c2-3) /(c-1)), where a, b, c are different from 1 , lie on the line l x+m y+n=0, then (A) a+b+c=-(m/l) (B) a b+b c+c a=(n/l) (C) a b c=((m+n)/l) (D) a b c-(b c+c a+a b)+3(a+b+c)=0

Tardigrade Admin · Accepted Answer

Since the given points lie on the line l x+m y+n=0, a, b, c are the roots of the equation l((t3/t-1))+m((t2-3/t-1))+n=0 or l t3+m t2+n t-(3 m+n)=0 Hence, a+b+c=-(m/l) a b+b c+c a=(n/l) a b c=(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

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

$a + b + c = - \frac{m}{l}$

$ab + b c + c a = \frac{n}{l}$

$ab c = \frac{( m + n )}{l}$

$ab c - (b c + c a + ab) + 3 (a + b + c) = 0$

Q. If the points (a3/(a−1),(a2−3)/(a−1)),(b3/(b−1), (b2−3)/(b−1)), and (c3/(c−1),(c2−3)/(c−1)), where a,b,c are different from 1 , lie on the line lx+my+n=0, then

a+b+c=−lm​

ab+bc+ca=ln​

abc=l(m+n)​

abc−(bc+ca+ab)+3(a+b+c)=0

Since the given points lie on the line lx+my+n=0,a,b,c are the roots of the equation l(t−1t3​)+m(t−1t2−3​)+n=0 orlt3+mt2+nt−(3m+n)=0 Hence, a+b+c=−lm​ ab+bc+ca=ln​ abc=l3m+n​ So, from (1), (2), and (3), we get abc−(bc+ca+ab)+3(a+b+c)=0

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

$a + b + c = - \frac{m}{l}$

$ab + b c + c a = \frac{n}{l}$

$ab c = \frac{( m + n )}{l}$

$ab c - (b c + c a + ab) + 3 (a + b + c) = 0$