First slide

Pair of straight lines

Question

The combined equation of straight lines passing through point $(- 1, 2)$ and each line makes intercepts on the coordinate axes whose product is

Moderate

A

$(x + y - 1) (4 x + y + 2) = 0$
B

$(x + y + 1) (4 x + y + 2) = 0$
C

$(x + y - 1) (4 x + y - 2) = 0$
D

$(x - y - 1) (4 x + y + 2) = 0$

Solution

The equation of any line which makes x- intercept a and y- intercept b is $\frac{x}{a} + \frac{y}{b} = 1$
Since the product of intercepts is equal to 1, one is the reciprocal of the other.
It means $a, \frac{1}{a}$ are intercepts of the line.
Hence, the equation of the line can be written as
$\frac{x}{a} + ay = 1 . . . . . . (1) It is passing through the point (- 1, 2) \begin{matrix} - \frac{1}{a} + 2 a = 1 \\ 2 a^{2} - a - 1 = 0 \end{matrix} The roots of the equation are a = 1, a = - \frac{1}{2} Substitute a = 1 in equation (1) It gives x + y - 1 = 0 Substitute a = - \frac{1}{2} in equation (1) It gives \begin{matrix} - 2 x - \frac{1}{2} y = 1 \\ - 4 x - y = 2 \\ 4 x + y + 2 = 0 \end{matrix}$
The combined equation of two lines is the product of both equations
Therefore, the combined equation of pair of lines is $(x + y - 1) (4 x + y + 2) = 0$

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