First slide

Equation of line in Straight lines

Question

If $\frac{1}{a}, \frac{1}{b}, \frac{1}{c}$ are in A.P., then the straight line $\frac{x}{a} + \frac{y}{b} + \frac{1}{c} = 0$ always passes through a fixed point, that point is

Moderate

A

(-1, -2)
B

(-1, 2)
C

(1, -2)
D

$(1, - \frac{1}{2})$

Solution

Since $\frac{1}{a}, \frac{1}{b}, \frac{1}{c}$ are in A.P.
$\Rightarrow \frac{1}{a} + \frac{1}{c} = \frac{2}{b}$ -----------(1)
The given line is $\frac{x}{a} + \frac{y}{b} + \frac{1}{c} = 0$
$\Rightarrow \frac{x}{a} + \frac{y}{b} + (\frac{2}{b} - \frac{1}{a}) = 0$ [Using (1)]
$\Rightarrow \frac{1}{a} (x - 1) + \frac{1}{a} (y + 2) = 0$
$\Rightarrow$ The given line passes through the point of intersection of x – 1 = 0 and y + 2 = 0 i.e., (1, –2) which is a fixed point.

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