A straight line cuts off the intercepts OA = a and OB = b on the positive directions of x -axis and y − axis respectively. If the perpendicular from origin O to this line makes an angle of 6 π with

Question

A straight line cuts off the intercepts OA = a and OB = b on the positive directions of x -axis and y − axis respectively. If the perpendicular from origin O to this line makes an angle of 6 π ​ with positive direction of y -axis and the area of △ OAB is 3 98 ​ 3 ​ , then a 2 − b 2 is equal to:

TestHub · Accepted Answer

Plotting the diagram of the given data we get,

Equation of straight line

Here given $O A = a & O B = b$ , so equation of line $A B$ in intercept form will be $\frac{x}{a} + \frac{y}{b} = 1 . . . . . . (1)$

And given perpendicular line of length $p$ is making $\frac{π}{3}$ angle with the $x -$ axis,

So, the equation of line in normal form will be

$x \cos (\frac{π}{3}) + y \sin (\frac{π}{3}) = p$

$\Rightarrow \frac{x}{2 p} + \frac{y \sqrt{3}}{2 p} = 1$

So, on comparing with the equation $(1)$ we get,

$a = 2 p & b = \frac{2 p}{\sqrt{3}}$

Now area of the triangle $O A B$ is

$\frac{1}{2} \times a \times b = \frac{98 \sqrt{3}}{3}$

$\Rightarrow \frac{1}{2} \times 2 p \times \frac{2 p}{\sqrt{3}} = \frac{98 \sqrt{3}}{3}$

$\Rightarrow p^{2} = 49$

Therefore,

$a^{2} - b^{2} = 4 p^{2} - \frac{4 p^{2}}{3} = \frac{8}{3} p^{2}$

$\Rightarrow a^{2} - b^{2} = \frac{8}{3} (49) = \frac{392}{3}$

Mathematics - Straight Lines Question with Solution | TestHub

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