Mathematics

3D Geometry and Coordinate Distance

140 Questions

Three dimensional geometry and coordinate distance problems involve calculating spatial measurements between points and lines. Questions feature vector distances, perpendicular distances, and coordinate geometry theorems. This advanced topic is typically found in mathematics examinations.

Point distance calculationsPerpendicular distancesVector coordinatesLine ratiosAxis distances

3D Geometry and Coordinate Distance Questions

Multiple choice maths fundamentals pair of straight lines distances and midpoints distance formula in 2d

Find $a$ if the distance between $(a , 2)$ and $(3 , 4)$  is $8 $

  1. $ 3 \, \pm \, \sqrt {60}$
  2. $ 4 \, \pm \, \sqrt {60}$
  3. $ 3 \, \pm \, \sqrt {6}$

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

By square of distance formual:


$8^2=(3-a)^2+(4-2)^2$

$=>64=a^2-6a+13$

$=>a^2-6a-51=0$

solving the quadratic we get:

$a=(3+\sqrt(60)$ or $(3-\sqrt(60))$.

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

Perpendicular distance between line $2x + y  =5,  2x + y  =3$ 

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

The given lines are


$2x+y=3\cdots(1)$

$2x+y=5\cdots(2)$

The perpendicular distance between lines is given as 

$\dfrac{|c _1-c _2|}{\sqrt {a^2+b^2}}$

$\dfrac{|5-3|}{\sqrt{ 2^2+1^2}}$

$\dfrac{2}{\sqrt 5}$

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

The distance between the lines given by $(x+7y)^{2}+4 \sqrt{2}(x+7y)-42=0,$ is

  1. $\displaystyle \frac{4}{5}$
  2. $4\sqrt{2}$
  3. $2$
  4. $10\sqrt{2}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Given equation of pair of straight lines is 

$(x+7y)^{2}+4 \sqrt{2}(x+7y)-42=0$

$\Rightarrow x^{2}+49y^{2}+14xy+4\sqrt{2}x+28\sqrt{2}y-42=0$

Here $a=1, b=49, h=7, g=2\sqrt{2},f=14\sqrt{2},c=-42$
Here $h^{2}=ab$

The given equation represents a pair of parallel lines.

$\displaystyle d=2\sqrt{\frac{g^{2}-ac}{a(a+b)}}$

$\Rightarrow d=2$

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

Distance between two lines respresented by the line pair, $x^2 -4xy + 4y^2 + x -2y -6 = 0$ is

  1. $\displaystyle \frac {1}{\sqrt 5}$
  2. $\sqrt 5$
  3. $2\sqrt 5$

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

As ${ h }^{ 2 }=4=1\times 4=ab$ and $\displaystyle b{ g }^{ 2 }=4\times \left( \frac { 1 }{ 4 }  \right) =1\times \left( 1 \right) =a{ f }^{ 2 }$
Therefore the lines in ${ x }^{ 2 }-4xy+4{ y }^{ 2 }+x-2y-6=0$ are parallel
and the distance between the two lines
$\displaystyle =\frac { 2\sqrt { { g }^{ 2 }-ac }  }{ \sqrt { a\left( a+b \right)  }  } =\frac { 2\sqrt { \dfrac { 1 }{ 4 } -1\times \left( -6 \right)  }  }{ \sqrt { 1\left( 1+4 \right)  }  } =\frac { \sqrt { 25 }  }{ \sqrt { 5 }  } =\sqrt { 5 } $

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

If the distance between the pair of parallel lines ${x}^{2}+2xy+{y}^{2}-8ax-8ay-9{a}^{2}=0$ is $25\sqrt {2}$, then $a$ is

  1. $ \pm4$
  2. $\pm 2$
  3. $\pm 3$
  4. $\pm 5$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The equation $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ represents the general equation of pair of lines which are parallel to each other, then the distance between them is given by,


$d = \left|2\sqrt{\dfrac{g^2 - ac}{a(a+b)}} \right|$ (or)  $d = \left|2\sqrt{\dfrac{f^2 - ac}{b(a+b)}} \right|$

Here, the equation is, 
  $x^2 + 2xy + y^2 - 8ax - 8ay - 9a^2 = 0$


i.e., $a = 1, b = 1, c = -9a^2, h = 1, f = -4a, g = -4a $

$\because d = 25\sqrt{2}$
  

$\implies 25\sqrt{2} = \left|2\sqrt{\dfrac{(-4a)^2 - 1(-9a^2)}{1(1+1)}} \right|$

$\implies 25\sqrt{2} = \left|2\sqrt{\dfrac{16a^2 + 9a^2}{2}} \right|$

$\implies 25\sqrt{2} = \sqrt{2} (5a) $

$\therefore a = \pm 5$ (Ans)

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

The distance between the two lines represented by the equation $9x^2 - 24 xy + 16y^2 - 12 x + 16y - 12 = 0$ is

  1. $\dfrac{8}{5}$
  2. $\dfrac{6}{5}$
  3. $\dfrac{11}{5}$

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

The above given equation
$9x^2-24xy+16y^2-12x+16y-12=0$ can be factorized as
$(3x-4y+2)(3x-4y-6)=0$
Since both the lines are parallel the distance between them will be
$d=\left|\dfrac{C _{2}-C _{1}}{\sqrt{3^2+4^2}}\right|$
$=\left|\dfrac{2-(-6)}{5}\right|$
$=\dfrac{8}{5}$

Multiple choice position of point wrt ellipse ellipse maths

The point at shortest distance from the line x+y=7 and lying on an ellipse $x^2 + 2y^2 =6$, has coordinates

  1. ($\sqrt{2}, \sqrt{2}$)
  2. ($0, \sqrt{3}$)
  3. ($\sqrt{5}, \dfrac{1}{\sqrt{2}}$)

  4. (2, 1)
Reveal answer Fill a bubble to check yourself
A Correct answer
Multiple choice maths operations adding and subtracting numbers using place value addition & subtraction mental additions and subtractions

The distance between (-4, -5) and (-4, -10) is________units

  1. 15

  2. 10

  3. 5

  4. 2
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Distance between two points $ ({x} _{1}, {y} _{1}) $ and $ ({x} _{2}, {y} _{2}) $ is $ \sqrt {{({x} _{2}-{x} _{1})}^{2} + {{(y} _{2}-{y} _{1})}^{2} } $

So, distance between $ (-4,-5) ; (-4,-10) $ is $ \sqrt {{(-4+4)}^{2} + {(-10+5)}^{2} } = 5 $

Multiple choice

How can the hyperfocal distance be calculated?

  1. $$H = rac{f^2}{Nc}$$
  2. $$H = rac{f}{Nc}$$
  3. $$H = rac{f^2}{N}$$
  4. $$H = rac{f}{N}$$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The hyperfocal distance (H) can be calculated using the formula $$H = rac{f^2}{Nc}$$, where f is the focal length of the lens, N is the aperture value (f-number), and c is the circle of confusion (typically 0.025mm for full-frame cameras).

Multiple choice

Let (X, d) be a metric space. Which of the following statements is true?

  1. The distance between any two points in X is always positive.

  2. The distance between any two points in X is always non-negative.

  3. The distance between any two points in X is always zero.

  4. The distance between any two points in X is always negative.
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

In a metric space, the distance between any two points is always non-negative. This is one of the fundamental properties of a metric space.

Multiple choice

Let (X, d) be a metric space. Which of the following is true about the distance between two sets A and B in X?

  1. The distance between A and B is the infimum of the distances between all pairs of points in A and B.

  2. The distance between A and B is the supremum of the distances between all pairs of points in A and B.

  3. The distance between A and B is the average of the distances between all pairs of points in A and B.

  4. The distance between A and B is the median of the distances between all pairs of points in A and B.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The distance between two sets A and B in a metric space (X, d) is the infimum of the distances between all pairs of points in A and B.