Mathematics

Straight Lines and Angles

162 Questions

Straight lines and angles are core components of coordinate geometry. This topic evaluates angle measures between intersecting lines, direction ratios, and perpendicular distances. Mastery of these mathematical concepts is necessary for high scores in quantitative exams.

Angle between linesDirection ratiosAngle bisectorsPerpendicular distanceSlope differences

Straight Lines and Angles 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

If the line $AX+BY=1$ passes through point of intersection of $y=x\tan\alpha+p\sec\alpha$,$y\sin(30-\alpha)-x\cos(30^ {o}-\alpha)=p$ and is inclined at $30^ {o}$ with $y=(x\tan\alpha+p\sec\alpha)$ then the value of $a^ {2}+b^ {2}=?$

  1. $\dfrac {1}{p^ {2}}$
  2. $\dfrac {2}{p^ {2}}$
  3. $\dfrac {3}{2p^ {2}}$
  4. $\dfrac {3}{4p^ {2}}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The intersection point of the given lines involves parameters p and alpha. By calculating the intersection and applying the condition of inclination, the sum of squares of coefficients A and B simplifies to 1/p^2.

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 the line passing through origin and making an angle $30^{\circ}$ with xaxis is

  1. $ x=\sqrt 3y$
  2. $ y=\sqrt 3x $
  3. $ x=3y$
  4. $ y=3x $
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The angle made by line is $30^{\circ}$


The slope of line is $m=\tan 30=\dfrac 1{\sqrt 3}$


Equation of line is $y=mx+c$

$y=\dfrac 1{\sqrt 3} x+c$

Put $(x,y)=(0,0)\Rightarrow c=0$

$\Rightarrow y=\dfrac 1{\sqrt 3} x$

$\therefore\ x=\sqrt 3 y$

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 intercepts made by a line on the co-ordinate axes are in the ratio $3:4$

and passes through $(3,0)$

  1. $3x+4y=12$
  2. $4x+3y=12$
  3. $4x-3y+12=0$

  4. None.
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let the common factor be $k$ 


The ratio is $3:4$


The intersepts are $3k,4k$

The equation is 
$\dfrac x{3k}+\dfrac y{4k}=1\\4kx+3ky=12k^2\\4x+3y=12k$

It passes through $(3,0)$

$\implies 4(3)+3(0)=12k\\k=1$

So the equation is $4x+3y=12$

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

If a line passes through the point $P(1, 2)$ makes an angle of $45^o$ with the x-axis and meets the line $x+2y-7=0$ in Q, then PQ equals?

  1. $\dfrac{2\sqrt{2}}{3}$
  2. $\dfrac{3\sqrt{2}}{2}$
  3. $\sqrt{3}$
  4. $\sqrt{2}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The line through (1, 2) with angle 45 degrees is y - 2 = 1(x - 1), or y = x + 1. Intersection with x + 2y - 7 = 0: x + 2(x + 1) - 7 = 0 => 3x = 5 => x = 5/3, y = 8/3. Distance PQ = sqrt((5/3 - 1)^2 + (8/3 - 2)^2) = sqrt((2/3)^2 + (2/3)^2) = sqrt(8/9) = 2*sqrt(2)/3.

Multiple choice maths linear graphs quadrants the cartesian plane the cartesian system

The acute angle between the lines $x-y=0$ and $y=0$ is

  1. $30^{\circ}$
  2. $45^{\circ}$
  3. $60^{\circ}$
  4. $75^{\circ}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Given 


$x-y=0$

$y=0$ represents $x-axis$ 

So the slope of line $x-y=0$ is $-\dfrac 1{-1}=1$

$\implies \tan \theta =1$

$\tan \theta =\tan \dfrac \pi 4$

$\theta =\dfrac \pi 4$

$\theta =45^{\circ}$

Multiple choice maths construction of angles identifying angles acute and obtuse angles types of angle

What is the acute angle between the lines y = 3x + 2 and y = 4x + 9?

  1. 4.4$\displaystyle ^{\circ}$
  2. 28.3$\displaystyle ^{\circ}$
  3. 5.2$\displaystyle ^{\circ}$
  4. 18.6$\displaystyle ^{\circ}$

  5. none of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$y=3x+2$

$m _1=3$
$=>tan\theta _1=3$
$=>\theta _1=tan^{-1}(3)$
$=>\theta _1=71.57^{0}$
Again,
$y=4x+9$
$=>m _2=4$
$=>tan\theta _2=4$
$=>\theta _2=tan^{1}(4)$
$=>\theta _2=75.96^{0}$
$\therefore$ Angle between the lines
$=\theta _2-\theta _1$
$=75.96-71.56$
$=4.4^{0}$

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

If the straight line through the point $P(3,4)$ makes an angle $\cfrac{\pi}{6}$ with the x-axis and meets the line $3x+5y+1=0$ at $Q$, the length $PQ$ is

  1. $\dfrac {132}{12\sqrt {3}+5}$
  2. $\dfrac {132}{12\sqrt {3}-5}$
  3. $\dfrac {132}{5\sqrt {3}+12}$
  4. $\dfrac {132}{5\sqrt {3}-12}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The equation of straight line passing through $P(3,4)$ is $y=\tan \dfrac{\pi}{6}{x}+(4-3\tan \dfrac{\pi}{6})\implies y=\dfrac{x}{\sqrt{3}}+4-\sqrt{3}$

The point of intersection will be $\bigg(\dfrac{55-57\sqrt{3}}{5+3\sqrt{3}},\dfrac{-10+3\sqrt{3}}{5+3\sqrt{3}}\bigg)$
Length will be $30(5-3\sqrt{3})$

Multiple choice angle between a line and a plane three dimensional geometry - ii product of vectors applications of vector algebra maths

The angle between the line $\dfrac{x-1}{1}=\dfrac{y+2}{1}=\dfrac{z-4}{0}$ and the plane $y+z+2=0$ is

  1. $\dfrac{\pi}{3}$
  2. $\dfrac{\pi}{4}$
  3. $\dfrac{\pi}{6}$
  4. $\dfrac{\pi}{2}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$\text cos\theta = \dfrac{1.0+1.1+0.1}{(√(1^{2}+1^{2}+0^{2}).(√0^{2}+1^{2}+1^{2})}$

$\text cos\theta = \dfrac{1}{√2.√2}$
$\text cos\theta = \dfrac{1}{2}$
$\theta = \dfrac{π}{3}$

Multiple choice angle between a line and a plane three dimensional geometry - ii product of vectors applications of vector algebra maths

The angle between the line $\dfrac{x}{2} = \dfrac{y}{3} = \dfrac{z}{4}$ and the plane $3x + 2y - 3z = 4$, is

  1. $45^o$
  2. $0^o$
  3. $\cos^{-1} \left(\dfrac{24}{\sqrt{29 \times 22}}\right)$
  4. $90^o$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Line$:\cfrac { x }{ 2 } =\cfrac { y }{ 3 } =\cfrac { z }{ 4 } $ has directions $(2,3,4)$

Plane$:3x+2y-3z=4$ has normal with direction ratios $(3,2,-3)$
$\therefore$ angle between plane and line be $\theta$ then angle between line and its direction will be $90-\theta$.
$\cos { (90-\theta ) } =\cfrac { 2\times 3+3\times 2+4\times -3 }{ \sqrt { ({ 2 }^{ 2 }+{ 3 }^{ 2 }+{ 4 }^{ 2 })({ 3 }^{ 2 }+{ 2 }^{ 2 }+{ 3 }^{ 2 }) }  } =0\ \therefore 90-\theta =90\quad \Rightarrow \theta ={ 0 }^{ \circ  }$

Multiple choice angle between a line and a plane three dimensional geometry - ii product of vectors applications of vector algebra maths

If $\overline {c}$ is perpendicular to $\overline {a}$ and $\overline {b}$ , $\left| \overline {a} \right| =3,\ \left| \overline {b} \right|=4,\ \left| \overline {c} \right|=5$ and the angle between $\overline {a}$ and $\overline {b}$ is $\dfrac{\pi}{6}$ then $[\overline {a}\ \ \ \overline {b}\ \ \ \overline {c}]=$

  1. $30\sqrt{3}$
  2. $30$
  3. $15$
  4. $15\sqrt{3}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

We have,

$\begin{matrix} \left[ { \overline { a } \, \, \overline { b } \, \, \overline { c }  } \right] =\overline { c } \times \left( { \overline { a } \times \overline { b }  } \right)  \ =\overline { c } \times \left( { \overline { a } \times \overline { b }  } \right) \cos { 0^{ 0 } }  \ =5\times 3\times 4\times \sin  \frac { \pi  }{ 6 }  \  \end{matrix}$
$ = 5 \times 3 \times 4 \times \frac{1}{2}$
$ = 30$
Then,
Option $B$ is correct answer.