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 what is meant by the equation of a straight line or of a curve introduction to slope introduction to straight lines
Find the equation of the straight line whose $x$ and $y$-intercepts on the axes are given by $2$ and $3$.
  1. $3x-2y+6=0$
  2. $3x+2y-6=0$
  3. $3x-2y-6=0$

  4. none of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Consider the given points.

$(2, 0)$ and $(0, 3)$

We know that the equation of the line which is passing through the points
$y-y _1=\dfrac{y _2-y _1}{x _2-x _1}(x-x _1)$

So,

$y-0=\dfrac{3-0}{0-2}(x-2)$

$y=\dfrac{3}{-2}(x-2)$

$-2y=3x-6$

$3x+2y-6=0$

Hence, Option $B$ is the answer.

Multiple choice mathematics and statistics coordinates, points and lines what is meant by the equation of a straight line or of a curve introduction to slope introduction to straight lines

The equation of line passing through $(5,4)$ and $(-9,3)$

  1. $x-14y+51=0 $
  2. $ x+y=2 $
  3. $ x-2 y=0 $
  4. $ 2 x-y=0 $
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
The equation of line Passing through $(5,4)$ and $(-9,3)$

The slope of line is $\dfrac{3-4}{5+9}=\dfrac 1{14}$

The equation of line is $y-4=\dfrac 1{14}(x-5)$

$\Rightarrow 14y-56=x-5$

$\Rightarrow x-14y+51=0$
Multiple choice mathematics and statistics coordinates, points and lines what is meant by the equation of a straight line or of a curve introduction to slope introduction to straight lines

Find the equation of a line which makes equal angles with the lines $x+y-2=0$ and $7x-y+4=0$ and passes through $(1, 2)$.

  1. $x-3y=5$
  2. $3x+y=5$
  3. $7x-y=5$
  4. $x+3y=7$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The angle bisectors of the lines x+y-2=0 and 7x-y+4=0 are found using the formula (x+y-2)/sqrt(2) = +/- (7x-y+4)/sqrt(50). Simplifying these and checking which passes through (1, 2) gives the equation.

Multiple choice maths banking and taxation reading graphs describing different situations using equations to plot lines basics of a straight line

If $m$ and $b$ are real numbers and $mb > 0$, then the line whose equation is $y = mx + b$ cannot contain the point-

  1. $(0, 2009)$
  2. $(2009, 0)$
  3. $(0, -2009)$
  4. $(20, -100)$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
$y= mx + b$
for (2009,0)
substituting in the given line
we get $2009m+b=0$
that is possible only if $mb < 0$
which contradicts our initial assumption mb > 0
so option is $b$
Multiple choice maths banking and taxation reading graphs describing different situations using equations to plot lines basics of a straight line

Consider the lines $2x+3y=0$,    $5x+4y=7$. Find the intersection point.

  1. $(3,-2)$
  2. $(3,2)$
  3. $(-3,2)$
  4. $(2,3)$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The lines are $2x+3y=0$......(1)


$x=\dfrac{-3y}{2}$

$5x+4y=7$........(2)

$5\left(\dfrac{-3y}{2}\right)+4y=7$

$-15y+8y=14$

$-7y=14$

$y=-2$

$x=\dfrac{-3(-2)}{2}=3$

$(x,y)=(3,-2)$

Multiple choice maths banking and taxation reading graphs describing different situations using equations to plot lines basics of a straight line

$y=2x+3$
Which of the following statements is true about the given line?

  1. The line passes through $(0,3)$ and $m=-2$
  2. The line passes through $(3,0)$ and $m=-2$
  3. The line passes through $(0,3)$ and $m=2$
  4. The line passes through $(3,0)$ and $m=2$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Comparing the given equation $y=2x+3$ with $y = mx+c$, we get

Hence, $m=2$ and $c=3$.
So, options A and B are incorrect.

Option C:
Substitute $(0,3)$ in the given equation, we get
RHS: $=2(0)+3 = 3$
LHS: $y=3$
$LHS =  RHS$, Hence option C is correct.

Option D:
Substitute $(3,0)$ in the given equation, we get
RHS: $=2(3)+3 = 9$
LHS: $y=0$

$LHS \neq RHS$. Hence, option D is incorrect.

Multiple choice maths banking and taxation reading graphs describing different situations using equations to plot lines basics of a straight line

Consider the equation of the line :$\displaystyle \frac{x-1}{3}-\frac{y+2}{2}=0$

  1. The line passes through $(4,0)$ and $m=2/3$
  2. The line passes through $(4,0)$ and $m=-2/3$
  3. The line passes through $(4,0)$ and $m=3/2$
  4. The line passes through $(4,0)$ and $m=-3/2$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
Given line 
$\dfrac{x-1}{3}-\dfrac{y+2}{2}=0$

$\dfrac{x-1}{3}=\dfrac{y+2}{2}$

$2(x-1)=3(y+2)$

$2x-2=3y+6$

$y=\dfrac{2x}{3}-\dfrac{8}{3}$

on comparing above eq with $y=mx+c$

$slope(m)=\dfrac{2}{3}$

y-intercept$=-\dfrac{8}{3}$

when $y=0,x=4$

Hence it passes through $(4,0)$ with $m=\dfrac{2}{3}$
Multiple choice maths banking and taxation reading graphs describing different situations using equations to plot lines basics of a straight line

Consider the line: y= -x +4

Which of the following is correct.

  1. The line passes through (0,4) and m=1.

  2. The line passes through (0,4) and m=-1.

  3. The line passes through (0,0) and m=-1.

  4. The line passes through (4,0) and m=-1.
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
Given line 
$y=-x+4$
on comparing above eq with $y=mx+c$
$slope(m)=-1$
y-intercept$=4$
Hence it passes through (0,4) with $m=-1$
Multiple choice maths banking and taxation reading graphs describing different situations using equations to plot lines basics of a straight line

Consider the equation of the line $\displaystyle x-3=\frac{2}{5}\left ( y-1 \right )$. Which of the following is correct?

  1. The line passes through $(6,5)$ and $m=-2/5$.
  2. The line passes through $(5,6)$ and $m=-5/2$.
  3. The line passes through $(6,5)$ and $m=2/5$.
  4. The line passes through $(5,6)$ and $m=5/2$.
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
Given line 
$x-3=\dfrac{2}{5}(y-1)$
$5(x-3)=2(y-1)$
$5x-15=2y-2$
$2y=5x-13$
$y=\dfrac{5x}{2}-\dfrac{13}{2}$
on comparing above eq with $y=mx+c$
$slope(m)=\dfrac{5}{2}$

when $x=5$
$2y=25-13$
$2y=12$
$y=6$
Hence it passes through (5,6) with $m=\dfrac{5}{2}$
Multiple choice maths banking and taxation reading graphs describing different situations using equations to plot lines basics of a straight line

For the pair of linear equations given below, draw graph and then state, whether the lines drawn are,
$\displaystyle y=3x-1$
$\displaystyle \frac{x}{2}+\frac{y}{3}=1$

  1. Perpendicular

  2. Parallel

  3. Intersecting but not at right angles

  4. Options B & C
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The first line is y = 3x - 1 (slope m1 = 3). The second line x/2 + y/3 = 1 can be rewritten as y = -3/2x + 3 (slope m2 = -3/2). Since m1 is not equal to m2 and their product is not -1, the lines intersect but are not perpendicular.

Multiple choice maths banking and taxation reading graphs describing different situations using equations to plot lines basics of a straight line

For the pair of linear equations given below, draw graphs and then state, whether the lines drawn are 
$\displaystyle 3x+4y=24$
$\displaystyle \frac{x}{4}+\frac{y}{3}=1$

  1. intersecting but not at right anglesl

  2. Options B & D

  3. perpendicular

  4. parallel
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The first line 3x + 4y = 24 has a slope of -3/4. The second line x/4 + y/3 = 1 can be rewritten as 3x + 4y = 12, which also has a slope of -3/4. Since the slopes are equal but the intercepts are different, the lines are parallel.

Multiple choice maths banking and taxation reading graphs describing different situations using equations to plot lines basics of a straight line

The straight lines given by the equations $\displaystyle x+y=2 , x-2y=5 \ and \ \frac{x}{3}+y=0$ are?

  1. concurrent

  2. intersecting to make a right triangle.

  3. intersecting to make an isosceles triangle.

  4. parallel to each other.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
Given lines
$x+y=2$------(1)
$x-2y=5$----(2) and $\dfrac{x}{3}+y=0$----(3)
Solving eq (1) and (2)
$x-2(2-x)=5$
$x-4+2x=5$
$x=3$ and $y=2-3=-1$
Point of intersection of line (1) and (2) is $P(3,-1)$
Solving eq (2) and (3)
$-3y-2y=5$
$-5y=5$
$y=-1$ and $x=-3y=3$
Point of intersection of line (2) and (3) is $Q(3,-1)$
Solving eq (1) and (3)
$-3y+y=2$
$-2y=2$
$y=-1$ and $x=-3y=3$
Point of intersection of line (1) and (3) is $R(3,-1)$
Here point of intersection of all line is same Hence line is concurrent
Multiple choice maths banking and taxation reading graphs describing different situations using equations to plot lines basics of a straight line

If the line ax + by + c = 0 is such that  a = 0 and b, $\displaystyle c\neq 0$ then the line is perpendicular to 

  1. x-axis

  2. y-axis

  3. x + y =1

  4. x = y
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

When $ a= 0 $ then the line equation becomes $ by + c = 0 $ or $ y = -\frac {c}{b} $

Equations of the form $ y =k $ are parallel to x-axis. This also means that they are perpendicular to $ y - $ axis as $ x-$ axis and $ y- $axis are perpendicular to each other.