Mathematics · Quantitative Aptitude

Coordinate Geometry

204 Questions

Practice fundamental coordinate geometry questions covering reflections, collinear points, and loci. This collection helps in mastering XY plane plots, midpoints, and quadrant identification. It is highly beneficial for students preparing for quantitative aptitude tests, JEE, and state level engineering exams.

Point reflectionCollinear pointsXY plane lociFinding midpointsCoordinate quadrantsRhombus vertices

Coordinate Geometry Questions

Multiple choice maths introduction to three dimensional geometry distance between two points in 3d distance between two points in space scalars and vectors

The coordinates of a point which is equidistant from the point $(0,0,0),(a,0,0),(0,b,0)$ and $(0,0,c)$ are given by

  1. $\displaystyle \left( \frac { a }{ 2 } ,\frac { b }{ 2 } ,\frac { c }{ 2 } \right) $
  2. $\displaystyle \left( \frac { -a }{ 2 } ,\frac { -b }{ 2 } ,\frac { c }{ 2 } \right) $
  3. $\displaystyle \left( \frac { a }{ 2 } ,\frac { -b }{ 2 } ,\frac { -c }{ 2 } \right) $
  4. $\displaystyle \left( \frac { -a }{ 2 } ,\frac { b }{ 2 } ,\frac { -c }{ 2 } \right) $
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let $P(x,y,z)$ be the required point.

Then $OP=PA=PB=PC$
Now $OP=PA \Longrightarrow OP^2+PA^2 \Longrightarrow x^2+y^2+z^2$

= $(x-a)^2+(y-0)^2+(z-0)^2 \Longrightarrow x=\dfrac{a}{2}$
Similarily, $OP=PB=y=\dfrac{b}{2}$ and $OP=PC=z=\dfrac{c}{2}$

hence the coordinate of the required point are $\left(\dfrac{a}{2},\dfrac{b}{2},\dfrac{c}{2}\right)$

Multiple choice maths constructions mid-point formula midpoints division of a line segment

The mid point of $(3,4)$ and $(1,-2)$

  1. (2,1)

  2. (1,2)

  3. (2,-1)

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

The points are $(3,4)$ and $(1,-2)$


The mid point of $(3,4)$ and $(1,-2)$ is given by 

$\left(\dfrac {x _1+x _2}2,\dfrac {y _1+y _2}2\right)\\\left(\dfrac{3+1}2,\dfrac {4-2}2\right)=(2,1)$

Multiple choice maths constructions mid-point formula midpoints division of a line segment

If (-2, -4) is the midpoint of (6, -7) and (x, y) then the values of x and y are

  1. x = 2, y = 1

  2. x = -10, y = -1

  3. x = 10, y = -1

  4. x = -8 , y = -1

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

Since, $(-2, -4)$ is the midpoint of $(6, -7)$ and $(x,y).$
$\Rightarrow \dfrac{x+6}2=-2\Rightarrow x=-10$
and $ \dfrac{y-7}2=-4\Rightarrow y=-1$
Option D is correct.

Multiple choice maths constructions mid-point formula midpoints division of a line segment

The coordinates of points $P(-2, 2), Q(3, 2) $ and $R(3, -2)$ are the vertices of a rectangle $PQRS$`. What are the coordinates of S? 

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

Mid-point of $PR=\left(\cfrac{-2+3}2,\cfrac{2-2}2\right)=\left(\cfrac12,0\right)$

Let coordinate of $S=(x,y)$
Mid-point of $QS=\left(\cfrac{3+x}2,\cfrac{2+y}2\right)$
Mid-point of $PR=$Mid-point of $QS$
$\Rightarrow\left(\cfrac12,0\right)=$$\left(\cfrac{3+x}2,\cfrac{2+y}2\right)$
We have $\cfrac12=\cfrac{3+x}2\Rightarrow x=-2$ and $0=\cfrac{2+y}2\Rightarrow y=-2$
Coordinate of $S=(-2,-2)$
Hence, B is the correct option.

Multiple choice maths constructions mid-point formula midpoints division of a line segment

If the mid-point between the points $(a+ b, a- b)$ and $(-a, b)$ lies on the line $ax + by = k$, what is k equal to?

  1. $\dfrac ab$
  2. $a + b$
  3. $ab$
  4. $a - b$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Mid point of points $(a+b,a-b),(-a,b)$ is $\left( \dfrac { a+b-a }{ 2 } ,\dfrac { a-b+b }{ 2 }  \right) =\left(\dfrac { b }{ 2 } ,\dfrac { a }{ 2 } \right)$

It lies on line $ax+by=k$ then 
$\Rightarrow a\times \dfrac{b}{2}+b\times \dfrac{a}{2}=k$
$\Rightarrow ab=k$

Multiple choice maths constructions mid-point formula midpoints division of a line segment

If $O(0,0)$ and $P(-8,0)$ then co-ordinates of its midpoint are________.

  1. $(-4,0)$
  2. $(4,0)$
  3. $(0,-4)$
  4. $(0,0)$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$O(0,0) $ and $P(-8,0)$ are given points.


The coordinates of the midpoint of 

$\overline{OP}=\left(\dfrac{x _1+x _2}{2},\dfrac{y _1+y _2}{2}\right)$

         $= \left( \dfrac {0-8}{2},\dfrac {0-0}{2}\right)$

         $=(-4,0)$

Multiple choice maths constructions mid-point formula midpoints division of a line segment

The point on $X-axis$ equidistant from $(2,3)$and $(1,5)$ is

  1. $\left( \dfrac { -13 }{ 2 } ,0 \right) $
  2. $\left( \dfrac { 13 }{ 2 } ,0 \right) $
  3. $(13,0)$
  4. $none\ of\ these$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let the point be (x, 0). Equidistance means (x-2)^2 + (0-3)^2 = (x-1)^2 + (0-5)^2. Expanding: x^2 - 4x + 4 + 9 = x^2 - 2x + 1 + 25. Simplifying: -4x + 13 = -2x + 26, so -2x = 13, x = -13/2. The point is (-13/2, 0).

Multiple choice maths constructions mid-point formula midpoints division of a line segment

The point (5,0) on y-axis is equidistant from (-1,2) and (3,4).

  1. True

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

A point on the y-axis has coordinates (0, y), but the question states the point is (5,0), which actually lies on the x-axis, creating a contradiction in the premise. Furthermore, calculating the distance from (5,0) to (-1,2) gives sqrt((5+1)^2 + (0-2)^2) = sqrt(36 + 4) = sqrt(40), and to (3,4) gives sqrt((5-3)^2 + (0-4)^2) = sqrt(4 + 16) = sqrt(20). Since these distances are not equal, the statement is false.

Multiple choice maths constructions mid-point formula midpoints division of a line segment

The point on $X$-axis which is equidistant from the point $\left( 3,5 \right )$ and $\left( 4,2 \right )$ is 

  1. $\left( -6,0 \right )$
  2. $\left( -7,0 \right )$
  3. $\left( 7,0 \right )$
  4. $\left( -5,0 \right )$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let the point on the X-axis be (x, 0). Since it is equidistant from (3,5) and (4,2), equate the squares of their distances: (x - 3)^2 + (0 - 5)^2 = (x - 4)^2 + (0 - 2)^2. Expanding this gives x^2 - 6x + 9 + 25 = x^2 - 8x + 16 + 4, which simplifies to 2x = -14, so x = -7. Thus, the point is (-7, 0).

Multiple choice maths constructions mid-point formula midpoints division of a line segment

Locus of the midpoints of the intercepts between the co-ordinate Axes by the lines passing through (a, 0) does not intersect

  1. X axis

  2. Y axis

  3. Y=x

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

Any line passing through (a, 0) has an intercept form x/p + y/q = 1 where it passes through (a, 0), meaning a/p + 0/q = 1, so p = a. The intercepts are (a, 0) and (0, q), so their midpoint has coordinates (h, k) = (a/2, q/2). This gives q = 2k, and since the y-intercept is q, the line intersects the y-axis at (0, 2k). However, the line cannot pass through the origin if it has distinct x and y intercepts, meaning q cannot be zero, which translates to the locus not intersecting the y-axis because h = a/2 is a vertical line parallel to the y-axis.

Multiple choice maths constructions mid-point formula midpoints division of a line segment

The line equally inclined to the coordinates axes and equidistant from points A(1, -2) and B(3, 4) is

  1. x+y=2, x+y=3

  2. x-y=3, x-y=1

  3. x-y=1, x+y=3

  4. x+y=2, x-y=3
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

A line equally inclined to the axes has a slope of either 1 or -1, meaning its equation is of the form x - y = c or x + y = c. Since it is equidistant from points A(1, -2) and B(3, 4), it must pass through the midpoint of AB or be parallel to the line segment AB. Testing the options reveals that the pair x - y = 1 and x + y = 3 satisfies the distance conditions from the midpoint and the slope requirements.

Multiple choice maths constructions mid-point formula midpoints division of a line segment

Find a point on the y-axis which equidistant from the points $A(6,5)$ and $B(-4,3)$

  1. $(0,9)$
  2. $(9,0)$
  3. $(3,0)$
  4. $(4,0)$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let the point be (0, y). Equidistance means the square of the distances to (6,5) and (-4,3) are equal: (0-6)^2 + (y-5)^2 = (0-(-4))^2 + (y-3)^2. This simplifies to 36 + y^2 - 10y + 25 = 16 + y^2 - 6y + 9, which results in 61 - 10y = 25 - 6y, or 4y = 36, so y = 9.