Mathematics

Straight Lines and Coordinates

155 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 maths position and movement reflection w.r.t a line transformation transformation and symmetry in geometrical shapes

The equation of the line AB is y = x. if And B lie on the same side of the line mirror 2x - y = 1, then the equation of the image of AB is   _____________.

  1. x + y - 2 = 0

  2. 8x + y - 9 = 0

  3. 7x - y - 6 = 0

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

Reflecting the line y=x across 2x-y=1 involves reflecting two points on the line (e.g., (0,0) and (1,1)) across the mirror line and finding the equation of the line passing through the images.

Multiple choice maths two dimensional analytical geometry pair of straight lines distances and midpoints distance formula in 2d

The equation of the line which passes through $(0,0)$ and $(1,1)$ is ____________

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

The equation of the line which passes through (0,0) and (1,1) is y=x

As we know, the two points are $(0,0) (1,1)$
slope $ m$ $=\dfrac { { y } _{ 2 }-{ y } _{ 1 } }{ { x } _{ 2 }-{ x } _{ 1 } } $  $=\dfrac { 1-0 }{ 1-0 } =1$
Standard equation of the line is $y-{ y } _{ 1 }=m\left( x-{ x } _{ 1 } \right) $
Substituting value of $m =1 $ and point $(x,y) =  (0,0)$ we get:
$y-0=1\left( x-0 \right) $
 $y=x$ is the required equation of the line

Multiple choice maths two dimensional analytical geometry pair of straight lines distances and midpoints distance formula in 2d

The equation of a straight line passing through points $(0,0)$ and $(1,5)$ is given by:

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

Given a line passes through $A(0,0)$ & $B(1,5)$ 

We know that the equation of line which passes through $(a _1,b _1)$ & $\left( { a } _{ 2 },{ b } _{ 2 } \right) $ is 
$y-{ b } _{ 1 }=\cfrac { { b } _{ 2 }-{ b } _{ 1 } }{ { a } _{ 2 }-{ a } _{ 1 } } (x-{ a } _{ 1 })\ \therefore \quad y-0=\cfrac { 5-0 }{ 1-0 } (x-0)\ y=5x$ 

Multiple choice maths two dimensional analytical geometry pair of straight lines distances and midpoints distance formula in 2d

Equation $4x^{2}+4xy-y^{2}-6x-3y-4=0$ represents a pair of parallel lines, then distance between these lines is

  1. $2\sqrt{5}$
  2. $\sqrt{5}$
  3. $\dfrac{2}{\sqrt{5}}$
  4. $\dfrac{3}{\sqrt{5}}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For the equation 4x^2 + 4xy - y^2 - 6x - 3y - 4 = 0, the distance between parallel lines is calculated using the formula 2 * sqrt(g^2 - ac) / sqrt(a(a+b)). Applying this to the given coefficients yields 2 * sqrt(5).

Multiple choice maths two dimensional analytical geometry pair of straight lines distances and midpoints distance formula in 2d

The four straight lines given by the equations $12x^2+7xy-12y^2=0$ and $12x^2+7xy-12y^2-x+7y-1=0$ lie along the sides of a 

  1. Square

  2. Rhombus

  3. Rectangle

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

taking $y$ is constant and finding the value of$x$ by roots formula.


$12x^2+(7y)x-12y^2=0$

$x=\dfrac{-b\pm \sqrt{b^2-4ac}}{2a}=\dfrac{(7y)-\sqrt{(7y)^2-4(-12)y^212}}{2\times 12}$

$\dfrac{\Rightarrow x=-7y\pm\sqrt{49y^2+576y^2}{}}{2\times 12}$

$\Rightarrow 24x=-7y\pm \sqrt{625}y$

$\Rightarrow 24xy=-7y\pm 25y$.........(1)

Again $12x^2+(7y-1)x+(-12y^2-1+7y)=0$

$\therefore x=\dfrac{-(7y-1)\pm\sqrt{(7y-1)^2-4}(12)(7y-1-12y^2)}{24}$

$\Rightarrow x=\dfrac{1-7y\pm\sqrt{49y^2+1-14y-336y+48+576y^2}}{24}$

$\Rightarrow 24x=1-7y\pm\sqrt{625y^2-350y+49}$

$\Rightarrow 24x=1-7y\pm (25y-7)$

$\Rightarrow x=\dfrac{1-7y\pm 25y-7}{24}$........(ii)

from (i) and (ii) we can clearly see co.efficient of $x$ and $y$ are same so slope are sample $m _1=\dfrac{24}{18},m _2=-\dfrac{24}{32}$

So $m _1m _2= -1$

$\therefore $ it is a square

Multiple choice maths two dimensional analytical geometry pair of straight lines distances and midpoints distance formula in 2d

$3x^2+8xy-3y^2=0$ represents a pair of lines AB and BC, whereas the equation $3x^2+8xy-3y^2+2x=4y-1=0$ represents two lines CD and DA.
Answer the given question.
The equation of the CD is,

  1. $x+3y+1=0$
  2. $x+3y-1=0$
  3. $x-3y+1=0$
  4. $x+y+1=0$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The equation 3x^2 + 8xy - 3y^2 + 2x - 4y - 1 = 0 represents a pair of lines. By factoring this quadratic, we can identify the individual linear equations for the sides CD and DA.

Multiple choice maths two dimensional analytical geometry pair of straight lines distances and midpoints distance formula in 2d

The value of $k$ so that the equation $12{x}^{2}-10{y}^{2}+11x-5y+k=0$ may represent a pair of straight lines is

  1. $k=\dfrac{91}{48}$
  2. $k=\dfrac{94}{43}$
  3. $k=\dfrac{83}{23}$

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

$\Rightarrow$  The given equation is $12x^2-10y^2+11x-5y+k=0$

$\Rightarrow$  Comparing it with $ax^2+2hxy+by^2+2gx+2fy+c=0$
$\Rightarrow$  $a=12,\,b=-10,\,h=o,\,g=\dfrac{11}{2},\,f=\dfrac{-5}{2},\,c=k$
$\Rightarrow$  The condition is $abc+2fgh-af^2-bg^2-ch^2=0$
$\Rightarrow$  $12\times (-10)\times k+2\times (\dfrac{-5}{2})\times\dfrac{11}{2}\times 0-12\times (\dfrac{-5}{2})^2-(-10)\times (\dfrac{11}{2})^2-k\times (0)^2=0$
$\Rightarrow$  $-120k+0-75+\dfrac{605}{2}-0=0$
$\Rightarrow$  $\dfrac{-240k-150+605}{2}=0$
$\Rightarrow$  $-240k+455=0$
$\Rightarrow$  $k=\dfrac{455}{240}$
$\therefore$   $k=\dfrac{91}{48}$

Multiple choice maths two dimensional analytical geometry pair of straight lines distances and midpoints distance formula in 2d

If the pair of lines ${x^2}\, + \,2xy\, + \,a{y^2}\, = \,0$ and $a{x^2}\, + \,2xy\, + \,{y^2}\, = \,0$ have exactly one line in common, then joint equation of the other two lines is given by

  1. $3{x^2}\, + \,8xy\, - 3\,{y^2}\, = \,0$
  2. $3{x^2}\, + \,10xy\, + 3\,{y^2}\, = \,0$
  3. ${y^2}\, + \,2xy\, - 3\,{x^2}\, = \,0$
  4. ${x^2}\, + \,2xy\, - 3\,{y^2}\, = \,0$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Given the two pairs of lines, we find the common line by solving the intersection of the quadratic forms. The joint equation of the remaining two lines is found by dividing the combined equation by the common factor.

Multiple choice maths two dimensional analytical geometry pair of straight lines distances and midpoints distance formula in 2d

Find the equations of the two straight lines drawn through the point $(0,a)$ on which the perpendicular let fall from the point $(2a,2a)$ are each of length $a$.
 then equation of the straight line joining the feet of these perpendiculars is $y+2x=5a$

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

By calculating the lines passing through (0, a) and applying the condition that the perpendicular distance from (2a, 2a) is 'a', we can verify the equation of the line joining the feet of the perpendiculars.

Multiple choice maths two dimensional analytical geometry pair of straight lines distances and midpoints distance formula in 2d

The line $x+3y-2=0$ bisects the angle between a pair of straight lines of which one has equation $x-7y+5=0$. The equation of the other line is-

  1. $3x+3y-1=0$
  2. $x-3y+2=0$
  3. $5x+5y-3=0$
  4. $none$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
we have
$L _1 =x+3y-2=0---(1)$

$L _2=x-7y+5=0---(2)$

now,

we know that

family of line through the given lines is

$L=L _2+\lambda L _2=0$

$=x-7y+5+\lambda (x+3y-2)=0---(3)$

Distance of any point ray $(2,0)$ on the line $x+3y-2=0$ from the lime 
$x-7y+5=0$ and the line $L=0$ must be same
so,

$\Rightarrow \ \left |\dfrac {2+5}{\sqrt {50}}\right | = \left |\dfrac {2+2\lambda +5-2\lambda}{\sqrt {(1+\lambda)^2+(3\lambda -7)^2}}\right|$

$\Rightarrow \ \dfrac {7}{\sqrt {50}}=\dfrac {7}{\sqrt {(1+\lambda)^2 +(3\lambda -7)^2}}$

$\Rightarrow \ 10\lambda^2-40\lambda =0$

$10\lambda (\lambda -4)=0$

$\lambda =0,\ \lambda =4$

then, put $\lambda =4$ in equation $(3)$ and we get

$L=x-7y+5+4(x+3y-2)=0$

$L=x-7y+5+4x+12y-8=0$

$L=5x+5y-3=0$

Hence this is the answer.


Multiple choice maths two dimensional analytical geometry pair of straight lines distances and midpoints distance formula in 2d

If the straight line $2x+3y+1=0$ bisects the  angle between a pair of lines ,one of which in this pair is $3x+2y+4=0$, then the equation of the other line in that pair of line is 

  1. $3x+4y-9=0$
  2. $6x-7y-14=0$
  3. $9x+46y-28=0$
  4. $9x-23y-12=0$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The bisector of the angle between two lines is given by the angle bisector theorem. Given one line and the bisector, the other line can be determined by reflecting the known line across the bisector.

Multiple choice maths two dimensional analytical geometry pair of straight lines distances and midpoints distance formula in 2d

In the equation $2x^{2}+2hxy+6y^{2}-4x+5y-6=0$ represent a pair of straight lines then the length of intercept on the $x-$axis cut by the lines is

  1. $2$
  2. $4$
  3. $\sqrt {7}$
  4. $0$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For a pair of lines ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0, the intercept on the x-axis is found by setting y=0 and solving the resulting quadratic in x. The distance between the roots is the intercept length.