If m=1 is the slope of a line L,then the prod | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Let the slope of the lines inclined at an angle of $60^{\circ}$ with line $L$ be $n$. Given the slope of line $L$ is $m=1$. The angle $	heta$ bet

Vedclass · Accepted Answer

$1$

Vedclass · Answer

(A) Let the slope of the lines inclined at an angle of $60^{\circ}$ with line $L$ be $n$. Given the slope of line $L$ is $m=1$. The angle $	heta$ between two lines with slopes $m$ and $n$ is given by $	an 	heta = \left| \frac{n-m}{1+nm} ight|$. Substituting the values,we have $	an 60^{\circ} = \left| \frac{n-1}{1+n} ight| = \sqrt{3}$. Squaring both sides,we get $\frac{(n-1)^2}{(n+1)^2} = 3$. $(n-1)^2 = 3(n+1)^2 $ $\Rightarrow n^2 - 2n + 1 = 3(n^2 + 2n + 1) $ $\Rightarrow n^2 - 2n + 1 = 3n^2 + 6n + 3 $ $\Rightarrow 2n^2 + 8n + 2 = 0 $ $\Rightarrow n^2 + 4n + 1 = 0$. The roots of this quadratic equation represent the slopes of the two lines. The product of the roots (slopes) is given by the constant term divided by the coefficient of $n^2$,which is $\frac{1}{1} = 1$.

If $m=1$ is the slope of a line $L$,then the product of the slopes of non-parallel lines which are inclined at an angle of $60^{\circ}$ with $L$ is

Similar Questions

The angle between the lines $\frac{x}{a} + \frac{y}{b} = 1$ and $\frac{x}{a} - \frac{y}{b} = 1$ is

Find the angle between the lines joining the points $(0, 0), (2, 3)$ and $(2, -2), (3, 5)$.

The value of $k (> 0)$,such that the angle between the lines $4x - y + 7 = 0$ and $kx - 5y - 9 = 0$ is $45^{\circ}$,is

The angle between the line $x+y=3$ and the line joining the points $(1,1)$ and $(-3,4)$ is

If $\theta$ is the acute angle between the lines $\frac{x}{a}+\frac{y}{b}=1$ and $\frac{x}{b}+\frac{y}{a}=1$,then $\sin \theta=$

Vedclass Test Series

Exam Paper Generator

Online Exam Module