Choose the correct answer. The tangent of angle between the lines… - Vidyadip

MCQ

Choose the correct answer. The tangent of angle between the lines whose intercepts on the axes are $a, -b$ and $b, -a,$ respectively, is

A
$\frac{\text{a}^2-\text{b}^2}{\text{ab}}$
B
$\frac{\text{b}^2-\text{a}^2}{2}$
✓
$\frac{\text{b}^2-\text{a}^2}{2\text{ab}}$
D
None of these.

✓

Answer

Correct option: C.

$\frac{\text{b}^2-\text{a}^2}{2\text{ab}}$

Intercepts of line are $a$ and $-b;$
i.e., line passes through the points $(a, 0), (0, -b).$
$\therefore$ Slope of line, $\text{m}_1=\frac{-\text{b}-0}{0-\text{a}}=\frac{\text{b}}{\text{a}}$
Intercepts of line are $b, -a;$
i.e., line passes through the points $(b, 0), (0, -a).$
$\therefore$ Slope of line, $\text{m}_2=\frac{-\text{a}-0}{0-\text{b}}=\frac{\text{a}}{\text{b}}$
If $\theta$ is the angle between the lines, then
$\tan=\theta=\frac{\frac{\text{b}}{\text{a}}-\frac{\text{a}}{\text{b}}}{1+\frac{\text{a}}{\text{b}}\times\frac{\text{b}}{\text{a}}}=\frac{\frac{\text{b}^2-\text{a}^2}{\text{ab}}}{2}=\frac{\text{b}^2-\text{a}^2}{2\text{ab}}$

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