Mathematics

Straight Lines and Coordinates

164 Questions

Straight lines and coordinates form the basis of coordinate geometry. This topic focuses on finding slopes, equations of lines, and points of intersection. These mathematical concepts are essential for performing well in advanced quantitative aptitude tests.

Line equations and slopesPoint of intersectionConcurrent linesNormal and parallel linesAngle between lines

Straight Lines and Coordinates Questions

Multiple choice mathematics and statistics coordinates, points and lines general equation of a line: ax+by+c=0 general equation of a line reducing equation of straight line to standard form

The slope and  the y-intercept  of the given line, $2x-3y = 7$ are respectively,

  1. $\dfrac{3}{2}, \dfrac{-3}{7}$
  2. $\dfrac{2}{3}, \dfrac{-7}{3}$
  3. $\dfrac{3}{2}, \dfrac{3}{7}$
  4. $\dfrac{2}{3}, \dfrac{7}{3}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The given equation is $2x-3y=7$ ........ $(1)$


To obtain the slope and $y-$intercept of the given equation, we write it in slope-intercept form which is, 

$y=mx+c$, where $m$ and $c$ are slope and $y-$intercept

From $(1)$,

$2x-3y=7\implies 3y=2x-7\implies y=\dfrac{2}{3} x+\left( -\dfrac{7}{3}  \right)$

Comparing it with $y=mx+c$ we get,

slope$=m=\dfrac{2}{3}$ and y-intercept$=c=-\dfrac{7}{3}$

Hence, option B is correct.

Multiple choice mathematics and statistics coordinates, points and lines general equation of a line: ax+by+c=0 general equation of a line reducing equation of straight line to standard form

The slope and y-intercept of the following line are respectively

$4x-y=0$

  1. $ slope=m=4\quad and\quad y-intercept=0.\\ $
  2. $ slope=m=-4\quad and\quad y-intercept=0.\\ $
  3. $ slope=m=1/4\quad and\quad y-intercept=0.\\ $
  4. $ slope=m=0\quad and\quad y-intercept=1/4.\\ $
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The given equation is $4x-y=0$ ........ $(1)$

To obtain the slope and $y-$intercept of the given equation, we write it in slope-intercept form which is 
$y=mx+c$, where $m$ and $c$ are slope and $y-$intercept

From $(1)$,
$4x-y=0\implies y=4x$
Comparing it with $y=mx+c$ we get,
slope$=m=4$ and y-intercept$=c=0$
Hence, option A is correct.

Multiple choice mathematics and statistics coordinates, points and lines general equation of a line: ax+by+c=0 general equation of a line reducing equation of straight line to standard form

The slope and y-intercept of the following line are respectively

$8x-4y-1=0$

  1. $ slope=m=\frac { -1 }{ 2 } \quad and\quad y-intercept=\frac { 1 }{ 4 } . $
  2. $ slope=m=2 \quad and\quad y-intercept=-\frac { 1 }{ 4 } . $
  3. $ slope=m=-\frac { 1 }{ 2 } \quad and\quad y-intercept=-\frac { 1 }{ 4 } . $
  4. $ slope=m=\frac { 1 }{ 2 } \quad and\quad y-intercept=\frac { 1 }{ 4 } . $
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
Given line
$8x-4y-1=0$
Comparing above eq with $y=mx+c$ where m is slope and c is y intercept
Here $m=2,c=-\dfrac{1}{4}$
Multiple choice mathematics and statistics coordinates, points and lines general equation of a line: ax+by+c=0 general equation of a line reducing equation of straight line to standard form

The slope and y-intercept of the following line are respectively

$5x - 2y = 3$

  1. $ slope=m=-\frac { 5 }{ 2 } \quad and\quad y-intercept=-\frac { 3 }{ 2 } . $
  2. $ slope=m=\frac { 5 }{ 2 } \quad and\quad y-intercept=\frac { 3 }{ 2 } . $
  3. $ slope=m=\frac { 5 }{ 2 } \quad and\quad y-intercept=-\frac { 3 }{ 2 } . $
  4. $ slope=m=-\frac { 5 }{ 2 } \quad and\quad y-intercept=\frac { 3 }{ 2 } . $
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The given equation is $5x-2y=3$ ........ $(1)$

To obtain the slope and $y-$intercept of the given equation, we write it in slope-intercept form which is 
$y=mx+c$, where $m$ and $c$ are slope and $y-$intercept

From $(1)$,
$5x-2y=3\implies y=\dfrac{5}{2}x - \dfrac{3}{2}$
Comparing it with $y=mx+c$ we get,
slope$=m=\dfrac{5}{2}$ and y-intercept$=c=-\dfrac{3}{2}$
Hence, option C is correct.

Multiple choice mathematics and statistics coordinates, points and lines general equation of a line: ax+by+c=0 general equation of a line reducing equation of straight line to standard form

The slope and $y$-intercept of the following line are respectively

$5x-8y =-2$

  1. slope $=m=-\dfrac { 5 }{ 8 } $ and $ y$-intercept $=\dfrac { 1 }{ 4 } $
  2. slope $=m=\dfrac { 5 }{ 8 } $ and $y$-intercept $=-\dfrac { 1 }{ 4 } $
  3. slope $=m=-\dfrac { 5 }{ 8 }$ and $ y$-intercept $=-\dfrac { 1 }{ 4 } $
  4. slope $=m=\dfrac { 5 }{ 8 } $ and $ y$-intercept $=\dfrac { 1 }{ 4 } $
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

$ To\quad obtain\quad the\quad slope\quad and\quad y-intercept\quad of\quad an\quad equation\quad of\quad any\quad form\ write\quad it\quad in\quad slope-intercept\quad form\quad which\quad is\quad y=mx+c.\ Then\quad slope=m\quad and\quad y-intercept=c.\  $
The given equation is: $5x - 8y = -2$
$-8y = -5x - 2$
$y = \dfrac{5}{8}x + \dfrac{2}{8} $
Compare it with general form of equation: $y = mx + c$
m = $\dfrac{5}{8}$, y - intercept = c = $  \dfrac{1}{4}$

Multiple choice mathematics and statistics coordinates, points and lines general equation of a line: ax+by+c=0 general equation of a line reducing equation of straight line to standard form

For the equation given below, find the the slope and the y-intercept : $\displaystyle 3y=7$

  1. $\displaystyle 0 \ and \ \frac{7}{3}$
  2. $\displaystyle 0 \ and \ -\frac{7}{3}$
  3. $\displaystyle -\frac{7}{3} \ and \ 0$
  4. $\displaystyle \frac{7}{3} \ and \ 0$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The equation of any straight line can be written as $ y =

mx + c $, where $m$ is its slope and $c$ is its y - intercept.
$ 3y = 7 $ can be written as $ y = \frac {7}{3} $

Comparing this equation with the standard form of the equation, we get:
$ m = 0 , c =  \frac {7}{3} $

Hence, slope of $ 3y = 7 $ is $ 0 $  and y -intercept is $   \frac {7}{3} $

Multiple choice mathematics and statistics coordinates, points and lines general equation of a line: ax+by+c=0 general equation of a line reducing equation of straight line to standard form

$ax + by + c = 0$ does not represent an equation of line if ____.

  1. $a = c = 0, b \neq 0$
  2. $b = c = 0, a \neq 0$
  3. $a = b = 0$
  4. $c = 0, a \neq 0, b \neq 0 $
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$ax+by+c=0$ will represent the equation of line If both or one coefficient of $x$ and $y$ is not equal to $0$.

Therefore, if $a=b=0$ then it will not represent the equation of a line.

Multiple choice mathematics and statistics coordinates, points and lines general equation of a line: ax+by+c=0 general equation of a line reducing equation of straight line to standard form

Find the slope and $y$-intercept of the line $2x + 5y = 1$

  1. slope $=$ $-\dfrac{2}{5}$, $y$-intercept $=$ $\dfrac{1}{5}$
  2. slope $=$ $-\dfrac{1}{5}$, $y$-intercept $=$ $\dfrac{1}{5}$
  3. slope $=$ $-\dfrac{2}{3}$, $y$-intercept $=$ $\dfrac{1}{5}$
  4. slope $=$ $-\dfrac{2}{5}$, $y$-intercept $= $ $\dfrac{2}{5}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
The slope intercept form of the line is $y=mx+c$, where $m$ is the slope of the line and $c$ is the $y$-intercept.
Change the equation $2x+5y=1$ in slope intercept form:
$2x+5y=1$
$5y=-2x+1$
$y=-\dfrac { 2 }{ 5 } x+\dfrac { 1 }{ 5 }$ 
Hence, the slope of the line $2x+2y=-2$ is $m=-\dfrac { 2 }{ 5 }$ and the $y$-intercept is $\dfrac { 1 }{ 5 }$.
Multiple choice mathematics and statistics coordinates, points and lines general equation of a line: ax+by+c=0 general equation of a line reducing equation of straight line to standard form

Find the slope and $y$-intercept of the line $-5x + y = 5$.

  1. slope $= 5, y$-intercept $= -5$
  2. slope $= 5, y$-intercept $= -4$
  3. slope $= 5, y$-intercept $= 5$
  4. slope $= 5, y$-intercept $= -1$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The slope-intercept form is $y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept. 

Given straight line $-5x+y=5$ can be written as, $y=5x+5$
Now comparing above equation with slope-intercept form $y=mx+c$

We get, slope $=m = 5$ and $y$-intercept $=c=5$.

Hence, option C is correct.

Multiple choice mathematics and statistics coordinates, points and lines general equation of a line: ax+by+c=0 general equation of a line reducing equation of straight line to standard form

Find the slope and $y$-intercept of the line $0.2x - y = 1.2$

  1. slope $= 0.2$, $y$-intercept $= -1.2$
  2. slope $= 1.2$, $y$-intercept $= -1.2$
  3. slope $= 0.2$, $y$-intercept $= -2.2$
  4. slope $= 0.2$, $y$-intercept $= -1.3$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
The slope intercept form of the line is $y=mx+c$, where $m$ is the slope of the line and $c$ is the $y$-intercept.
Change the equation $0.2x-y=1.2$ in slope intercept form:
$0.2x-y=1.2$
$\Rightarrow -y=-0.2x+1.2$
$\Rightarrow y=0.2x-1.2$
Hence, the slope of the line $0.2x-y=1.2$ is $m=0.2$ and the $y$-intercept is $-1.2$.
Multiple choice mathematics and statistics coordinates, points and lines general equation of a line: ax+by+c=0 general equation of a line reducing equation of straight line to standard form

Find the slope and $y$-intercept of the line $2x + 2y = -2$

  1. slope = 1, y-intercept $= -3$
  2. slope = -1, y-intercept $= -1$
  3. slope = 1, y-intercept $= 3$
  4. slope = 1, y-intercept $= 1$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
The slope intercept form of the line is $y=mx+c$, where $m$ is the slope of the line and $c$ is the $y$-intercept.
Change the equation $2x+2y=-2$ in slope intercept form:
$2x+2y=-2$
$\Rightarrow 2y=-2x-2$
$\Rightarrow y=-x-1$
Hence, the slope of the line $2x+2y=-2$ is $m=-1$ and the $y$-intercept is $-1$.
Multiple choice mathematics and statistics coordinates, points and lines general equation of a line: ax+by+c=0 general equation of a line reducing equation of straight line to standard form

Find the slope and $y$-intercept of the line $x - y = 3$

  1. slope $= 2$, $y$-intercept $= -3$
  2. slope $= 0$, $y$-intercept $= -3$
  3. slope $= 1$, $y$-intercept $= -3$
  4. slope $= 1$, $y$-intercept $= 3$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
The slope intercept form of the line is $y=mx+c$ where $m$ is the slope of the line and $c$ is the $y$-intercept.
Change the equation $x-y=3$ in slope intercept form:
$x-y=3$
$\Rightarrow -y=-x+3$
$\Rightarrow y=x-3$
Hence, the slope of the line $x-y=3$ is $m=1$ and the $y$-intercept is $-3$.
Multiple choice mathematics and statistics coordinates, points and lines general equation of a line: ax+by+c=0 general equation of a line reducing equation of straight line to standard form

Find an equation of the line through the points $(-3,5)$ and $(9,10)$ and write it in standard form $Ax+By=C$, with $A>0$

  1. $6x-10y=-75$
  2. $5x-12y=-75$
  3. $4x-11y=-65$
  4. $x-6y=-15$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
Given points are $(-3,5)$ and $(9,10)$
The slope of the line is given by:
$m =\dfrac{ (10-5)}{[9-(-3)] }= \dfrac {5}{12}$
The equation becomes: 
$y - 5 = \left (\dfrac {5}{12}\right) [x-(-3)]$ 
$y-5=\left (\dfrac {5}{12}\right)(x+3)$ 
Solve it and get the equation- 
$12y-60=5x+15$ 
$5x-12y=-75$ 

Multiple choice business maths pair of straight lines condition for perpendicular and coincident lines and bisectors of angles pair of straight lines through origin analytical geometry

Find the equation of the line perpendicular to $x-7y+5=0$ and having x-intercept 3.

  1. $x+y-21=0$
  2. $x-7y+21=0$
  3. $7x+y-21=0$

  4. None of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The equation of a line perpendicular to $x-7y+5=0$ is $7x+y+\lambda =0$.


Its x-intercept is 3. This means that the line cuts x-axis at a distance of 3 units from the origin. Consequently, it passes through the point (3,0) on x-axis.


Therefore, $21+0+\lambda =0$

$\lambda = -21$

Thus, the equation of the required line is $7x+y-21=0$.