First slide

Equation of line in Straight lines

Question

An equation of a straight line passing through the inter-section of the straight lines $3 x - 4 y + 1 = 0$ and $5 x + y - 1 = 0$ and making non-zero, equal intercepts on the axes is

Moderate

A

$22 x + 22 y = 13$
B

$23 x + 23 y = 11$
C

$11 x + 11 y = 23$
D

$8 x - 3 y = 0$

Solution

Equation of any line through the point of intersection of the given lines is
$(3 x - 4 y + 1) + k (5 x + y - 1) = 0$
or $(3 + 5 k) x + (k - 4) y + 1 - k = 0$
or $\frac{x}{(k - 1) / (3 + 5 k)} + \frac{y}{(k - 1) / (k - 4)} = 1$
Since $x$ -intercept = $y$ -intercept
$\Rightarrow \frac{k - 1}{3 + 5 k} = \frac{k - 1}{k - 4} \Rightarrow (k - 1) (3 + 5 k - k + 4) = 0$
$\Rightarrow k = 1 or k = - 7 / 4$
For $k$ = 1, (1) becomes 8x – 3y = 0 which makes zero intercepts on the axes.
$∴ k = - 7 / 4 \Rightarrow$ The required equation is
$\begin{array}{l} 4 (3 x - 4 y + 1) - 7 (5 x + y - 1) = 0 \\ \Rightarrow 23 x + 23 y = 11 . \end{array}$

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