Embibe Experts Solutions for Chapter: Straight Line, Exercise 1: Exercise 1

Consider the following points

$\mathrm{I}.$ $\left(0,5\right)$

$\mathrm{II}.$ $\left(2,-1\right)$

$\mathrm{III}.$ $\left(3,-4\right)$

Which of the above lie on the line $3x+y=5$ and at a distance $\sqrt{10}$ from $(1,2)$?

$(a,2b)$ is the midpoint of the line segment joining the points $\left(10,-6\right)$ and $(k,4).$ If $a-2b=7,$ then what is the value of $k?$

The angle between the lines $x+y-3=0$ and $x-y+3=0$ is $\alpha$ and the acute angle between the lines $x-\sqrt{3}y+2\sqrt{3}=0$ and $\sqrt{3}x-y+1=0$ is $\beta$. Which one of the following is correct?

The distance of the point $\left(1,3\right)$ from the line $2x+3y=6$, measured parallel to the line $4x+y=4$, is

The perpendicular distance between the straight lines $6x+8y+15=0$ and $3x+4y+9=0$ is

Consider the following statements
$1$. The distance between the lines $y=mx+c_1$ and $y=mx+c_2$ is
$\frac{\left|c_1-c_2\right|}{\sqrt{1+m^2}}$

$2$. The distance between the lines $ax+by+c_1=0$
and $ax+by+c_2=0$ is $\frac{\left|c_1-c_2\right|}{\sqrt{a^2+b^2}}.$.
$3$. The distance between the lines $x=c_1$ and $x=c_2$ is $\left|c_1-c_2\right|$

Which of the above statements are correct?

What is the angle between the straight lines $\left(m^2-mn\right)y=\left(mn+n^2\right)x+n^3$ and $\left(mn+m^2\right)y=\left(mn-n^2\right)x+m^3,$ where $m>n?$

The equation of the line joining the origin to the point of intersection of the lines $\frac{x}{a}+\frac{y}{b}=1$ and $\frac{x}{b}+\frac{y}{a}=1$ is