# A Line Passes Through a Point a (1, 2) and Makes an Angle of 60° with the X-axis and Intersects the Line X + Y = 6 at the Point P. Find Ap. - Mathematics

A line passes through a point A (1, 2) and makes an angle of 60° with the x-axis and intersects the line x + y = 6 at the point P. Find AP.

#### Solution

Here,

$\left( x_1 , y_1 \right) = A \left( 1, 2 \right), \theta = {60}^\circ$

So, the equation of the line is

$\frac{x - x_1}{cos\theta} = \frac{y - y_1}{sin\theta} = r$

$\Rightarrow \frac{x - 1}{\cos {60}^\circ} = \frac{y - 2}{\sin {60}^\circ} = r$

$\Rightarrow \frac{x - 1}{\frac{1}{2}} = \frac{y - 2}{\frac{\sqrt{3}}{2}} = r$

$\text { Here, r represents the distance of any point on this line from point } A (1, 2) .$

$\text { The coordinates of any point P on this line are } \left( 1 + \frac{r}{2}, 2 + \frac{\sqrt{3}r}{2} \right) .$

Clearly, P lies on the line x + y = 6

$\therefore 1 + \frac{r}{2} + 2 + \frac{\sqrt{3}r}{2} = 6$

$\Rightarrow \frac{\sqrt{3}r}{2} + \frac{r}{2} = 3$

$\Rightarrow r\left( \sqrt{3} + 1 \right) = 6$

$\Rightarrow r = \frac{6}{\sqrt{3} + 1} = 3\left( \sqrt{3} - 1 \right)$

∴ AP = $3\left( \sqrt{3} - 1 \right)$

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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 23 The straight lines
Exercise 23.8 | Q 1 | Page 65