Question

A member of these family with positive gradient making an angle of $\frac{\pi }{4}$ with the line 3x-4y=2, is

• A. 7 x - y - 5 = 0
• B. 4 x - 3 y +2 = 0
• C. x + 7 y = 15
• D. 5 x - 3 y - 4 = 0

Answer 1

The correct option is A 7 x - y - 5 = 0

According to the problem, we have

$y-2=m\left(x-1\right)....\left(eq.1\right)$

it makes angle $\frac{\pi }{4}$ with line and the slope is $\frac{3}{4}$

Since,

$\left|\frac{m-\left(\frac{3}{4}\right)}{1+\left(\frac{3m}{4}\right)}\right|=1$

implies that,

$\left|\frac{m-\left(\frac{3}{4}\right)}{1+\left(\frac{3m}{4}\right)}\right|=\pm 1$

With +ve sign

$4m-3=4+4m$

$m=7$

Equation of first line will be $\left(y-2\right)=7\left(x-1\right)$

For -ve slope

$4m-3=-4-3m$

$7m=-1$

$m=\frac{-1}{7}$

Hence, the line is $7x-y-5=0$

Therefore (A) will be true