First slide

Equation of line in Straight lines

Question

If three points $(h, 0), (a, b)$ and $(o, k)$ lie on a line then

Moderate

A

$a h + b k = 1$
B

$\frac{a}{h} + \frac{b}{k} = 1$
C

$a k + b h = 1$
D

$\frac{a}{k} + \frac{b}{h} = 1$

Solution

If given points lie on a line, then
$|\begin{array}{lll} h 0 1 \\ a b 1 \\ 0 k 1 \end{array}| = 0$
Expanding along first row
$\begin{array}{l} h (b - k) + 1 \cdot (a k - b \times 0) = 0 \\ \Rightarrow b h + a k = h k \\ \Rightarrow \frac{a}{h} + \frac{b}{k} = 1 \end{array}$
Alternatively: Equation of the line joining $(h, 0)$ and $(0, k)$ is $\frac{x}{h} + \frac{y}{k} = 1$ as it makes intercepts $h$ on $x -$ axis and $k$ on $y -$ axis
If the point $(a, b)$ also lies on it. then
$\frac{a}{h} + \frac{b}{k} = 1$

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