First slide

Equation of line in Straight lines

Question

If $x_{1}, x_{2}, x_{3}$ are the abscissa of the points $A_{1}, A_{2}, A_{3}$ respectively where the lines $y = m_{1} x, y = m_{2} x, y = m_{3} x$ meet the line $2 x - y + 3 = 0$ such that $m_{1}, m_{2}, m_{3}$ are in A.P., then $x_{1}, x_{2}, x_{3}$ , are in

Moderate

A

A.P.
B

G.P.
C

H.P.
D

none of these

Solution

We have $x_{1} = \frac{3}{m_{1} - 2}, x_{2} = \frac{3}{m_{2} - 2}, x_{3} = \frac{3}{m_{3} - 2}$
$\Rightarrow \frac{1}{x_{1}} + \frac{1}{x_{3}} = \frac{m_{1} - 2 + m_{3} - 2}{3} = \frac{m_{1} + m_{3} - 4}{3}$
$= \frac{2 m_{2} - 4}{3} = \frac{2}{x_{2}}$
(as $m_{1}, m_{2}, m_{3}$ are in A.P.)
$x_{1}, x_{2}, x_{3}$ are in H.P.

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