If the points a3a-1- a2 - 3a-1- b3b-1- b2 - 3b-1 and c3c-1-c2 -3c-1- where a- b- c are different from 1- lie on the line lx - my - n - 0- then

The correct options are A ab + bc + ca = nl B abc (bc+ca + ab) + 3 (a+b+c) = 0 C a+b+c = ml Since the given point lies on the line lx ...

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Byju's Answer

Standard XII

Mathematics

Graph of Quadratic Expression

If the points...

Question

If the points $(\frac{a^{3}}{a - 1}, \frac{a^{2} - 3}{a - 1}), (\frac{b^{3}}{b - 1}, \frac{b^{2} - 3}{b - 1})$ and $(\frac{c^{3}}{c - 1}, \frac{c^{2} - 3}{c - 1})$ , 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}

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

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a b c = \frac{(m + n)}{l}

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

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Solution

The correct options are
A $a b + b c + c a = \frac{n}{l}$
B $a b c - (b c + c a + a b) + 3 (a + b + c) = 0$
C $a + b + c = - \frac{m}{l}$
Since the given point lies on the line $l x + m y + n = 0$ , so $a, b, c$ are the roots of the equation
$l (\frac{t^{3}}{t - 1}) + m (\frac{t^{2} - 3}{t - 1}) + n = 0$ or
$l t^{3} + m t^{2} + n t - (3 m + n) = 0$
Hence, $a + b + c = - \frac{m}{l}$ ...... (i)
$a b + b c + c a = \frac{n}{l}$ ...... (ii)
$a b c = \frac{3 m + n}{l}$ ...... (iii)
So, from Equations (i), (ii) and (iii), we get
$a b c - (b c + c a + a b) + 3 (a + b + c) = 0$

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Similar questions

Q. If the points

(\frac{a^{3}}{a - 1}, \frac{a^{2} - 3}{a - 1}), (\frac{b^{3}}{b - 1}, \frac{b^{2} - 3}{b - 1}), (\frac{c^{3}}{c - 1}, \frac{c^{2} - 3}{c - 1})

are collinear for three distinct values of

a, b, c

, then show that

a b c - (b c + c a + a b) + 3 (a + b + c) = 0.

If a+b+c=0 then $(\frac{a^{2}}{b c} + \frac{b^{2}}{c a} + \frac{c^{2}}{a b})$ =?
(a) 1 (b) 0 (c) -1 (d) 3

If $(\frac{a^{3}}{a - 1}, \frac{a^{2} - 3}{a - 1}), (\frac{b^{3}}{b - 1}, \frac{b^{2} - 3}{b - 1}) and (\frac{c^{3}}{c - 1}, \frac{c^{2} - 3}{c - 1})$ are collinear and $α (a b c) + β (a + b + c) = γ (a b + b c + c a)$ , where $α, β, γ ϵ N$ , then find the least value of $α + β + γ$ . ___

Q. The expression

a^{3} + b^{3} + c^{3} - 3 a b c

can be expressed as a product of two expressions. What is the product?

Q. Prove that

\frac{a b}{c^{3}} + \frac{b c}{a^{3}} + \frac{c a}{b^{3}} > \frac{1}{a} + \frac{1}{b} + \frac{1}{c}

, where

a, b, c

are different positive real numbers.

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If the points (a3a−1,a2−3a−1),(b3b−1,b2−3b−1) and (c3c−1,c2−3c−1), where a,b,c are different from 1, lie on the line lx+my+n=0, then

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