Question

Two lines are drawn through $(3,4)$, each of which makes an angle of $45°$ with the line$x-y=2$, then the area of the triangle formed by these lines is

• A. $9$
• B. $\frac{9}{2}$
• C. $2$
• D. $\frac{2}{9}$

Answer 1

The correct option is B

$\frac{9}{2}$

Find the required area:

The three equations of lines,

$x=3.....\left(1\right)$

$y=4......\left(2\right)$

$x-y=2....\left(3\right)$

From $\left(1\right)$ and $\left(2\right)$

Coordinates of the point, $x=3$ and $y=4$

From $\left(2\right)$ and $\left(3\right)$

Coordinates of the point, $x=6$ and $y=2$

From $\left(3\right)$ and $\left(1\right)$

Coordinates of the point, $x=3$ and $y=1$

Area of the triangle formed by these lines $=\left|\frac{1}{2}\left(x_1\right(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2\left)\right)\right|$

$$\begin{gathered} =\left|\frac{1}{2}\left(3\right(2-1)+6(1-4)+3(4-2\left)\right)\right| \\ =\left|\frac{-9}{2}\right|=\frac{9}{2} \end{gathered}$$

Hence, the correct option is (B).