Name: Angle Between Two Lines
Uploaded: 2019-07-19
Duration: 6 min 13 s
Description: If two lines are perpendicular to each other, their direction vectors are also perpendicular. This means the scalar product of the direction vectors is equal...

Question

Find the equation of a straight line on which the perpendicular from the origin makes an angle of

30 °

with

x -

axis and which forms a triangle of area

50 / \sqrt{3}

with the axes.

Accepted Answer

Let $A B$ be the given line and $O L = p$ be the perpendicular drawn from the origin on the line.

Here, $α = 30 °$

So, the equation of the line $A B$ is

$x \cos α + y \sin α = p$

$\Rightarrow x \cos 30 ° + y \sin 30 ° = p$

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

$\Rightarrow \sqrt{3} x + y = 2 p . . . (1)$

Now, in triangles $O L A$ and $O L B$

$\cos 30 ° = \frac{O L}{O A} and \cos 60 ° = \frac{O L}{O B}$

$\Rightarrow \frac{\sqrt{3}}{2} = \frac{p}{O A} and \frac{1}{2} = \frac{p}{O B}$

$\Rightarrow O A = \frac{2 p}{\sqrt{3}} and O B = 2 p$

It is given that the area of triangle $O A B$ is $50 / \sqrt{3}$

$∴ \frac{1}{2} \times O A \times O B = \frac{50}{\sqrt{3}}$

$\Rightarrow \frac{1}{2} \times \frac{2 p}{\sqrt{3}} \times 2 p = \frac{50}{\sqrt{3}}$

$\Rightarrow p^{2} = 25$

$\Rightarrow p = 5$

Substituting the value of $p$ in $(1) :$
$\sqrt{3} x + y = 10$

Hence, the equation of the line $A B$ is $\sqrt{3} x + y = 10$ .

Find the equation of a straight line on which the perpendicular from the origin makes an angle of $30 °$ with $x -$ axis and which forms a triangle of area $50 / \sqrt{3}$ with the axes.

Important Questions on The Straight Lines

Learn and Prepare for any exam you want !

Find the equation of a straight line on which the perpendicular from the origin makes an angle of 30° with x-axis and which forms a triangle of area 50/3 with the axes.

Important Questions on The Straight Lines

Learn and Prepare for any exam you want !

Find the equation of a straight line on which the perpendicular from the origin makes an angle of $30 °$ with $x -$ axis and which forms a triangle of area $50 / \sqrt{3}$ with the axes.