The equations of the lines passing through th | vedclass

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

Question

(A) The equation of a line passing through $(1, 0)$ with slope $m$ is given by $y - 0 = m(x - 1)$,which simplifies to $mx - y - m = 0$.The perpendicul

Vedclass · Accepted Answer

$\sqrt{3}x + y - \sqrt{3} = 0, \sqrt{3}x - y - \sqrt{3} = 0$

Vedclass · Answer

(A) The equation of a line passing through $(1, 0)$ with slope $m$ is given by $y - 0 = m(x - 1)$,which simplifies to $mx - y - m = 0$.The perpendicular distance $d$ from the origin $(0, 0)$ to the line $Ax + By + C = 0$ is given by $d = \frac{|C|}{\sqrt{A^2 + B^2}}$.Here,$A = m, B = -1, C = -m$,and $d = \frac{\sqrt{3}}{2}$.Substituting these values: $\frac{|-m|}{\sqrt{m^2 + (-1)^2}} = \frac{\sqrt{3}}{2}$.Squaring both sides: $\frac{m^2}{m^2 + 1} = \frac{3}{4}$.$4m^2 = 3m^2 + 3 \implies m^2 = 3 \implies m = \pm \sqrt{3}$.For $m = \sqrt{3}$,the equation is $\sqrt{3}x - y - \sqrt{3} = 0$.For $m = -\sqrt{3}$,the equation is $-\sqrt{3}x - y + \sqrt{3} = 0$,which is $\sqrt{3}x + y - \sqrt{3} = 0$.

The equations of the lines passing through the point $(1, 0)$ and at a distance $\frac{\sqrt{3}}{2}$ from the origin are:

Similar Questions

Reduce the equation $x-y=4$ into the normal form $x \cos \omega + y \sin \omega = p$. Find the perpendicular distance from the origin $(p)$ and the angle between the perpendicular and the positive $x$-axis $(\omega)$.

The line $L$ is given by $\frac{x}{5} + \frac{y}{b} = 1$ and passes through the point $(13, 32)$. The line $K$ is parallel to $L$ and has the equation $\frac{x}{c} + \frac{y}{3} = 1$. Then the distance between $L$ and $K$ is

Let $A$ be the point of intersection of the lines $3x + 2y = 14$ and $5x - y = 6$. Let $B$ be the point of intersection of the lines $4x + 3y = 8$ and $6x + y = 5$. The distance of the point $P(5, -2)$ from the line $AB$ is:

$A$ point equidistant from the lines $4x + 3y + 10 = 0$,$5x - 12y + 26 = 0$ and $7x + 24y - 50 = 0$ is

The number of straight lines that can be drawn through the point $(-3, 4)$ which are at a distance of $5$ units from the point $(2, -8)$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module