First slide

Equation of line in Straight lines

Question

If the points $P_{i} = (\frac{a_{i}^{3}}{a_{i} - 1}, \frac{a_{i}^{2} - 3}{a_{i} - 1}), i = 1, 2, 3$ , , are collinear and a_1, a₂ and a₃ are distinct real numbers, then

Moderate

A

$a_{1} a_{2} a_{3} - \sum a_{1} a_{2} - 3 \sum a_{1} = 0$
B

$a_{1} a_{2} a_{3} + \sum a_{1} a_{2} - 3 \sum a_{1} = 0$
C

$a_{1} a_{2} a_{3} - \sum a_{1} a_{2} + 3 \sum a_{1} = 0$
D

$- a_{1} a_{2} a_{3} + \sum a_{1} a_{2} - 3 \sum a_{1} = 0$

Solution

Let the given points lie on the line lx + my + n = 0
$∴ (\frac{a_{i}^{3}}{a_{1} - 1}) + m (\frac{a_{i}^{2} - 3}{a_{i} - 1}) + n = 0$
$\Rightarrow l a_{i}^{3} + m a_{i}^{2} + n a_{i} - 3 m - n = 0$
$∴ \sum a_{1} = - \frac{m}{l}, \sum a_{1} a_{2} = \frac{n}{l} a n d a_{1} a_{2} a_{3} = \frac{3 m + n}{l}$
$∴$ $a_{1} a_{2} a_{3} - \sum a_{1} a_{2} + 3 \sum a_{1} = 0$

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