Mathematics

3D Geometry and Coordinate Distance

137 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 mathematics and statistics hyperbola parametric equation of the hyperbola forms of equations of a hyperbola equations of hyperbola

For hyperbola  $-\dfrac{(x-1)^2}{3}+\dfrac{(y+2)^2}{16}=1$ distance between directrices is ?

  1. $\dfrac{2}{\sqrt{19}}$
  2. $\dfrac{3}{\sqrt{19}}$
  3. $\dfrac{4}{\sqrt{19}}$
  4. $\dfrac{32}{\sqrt{19}}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
Comparing the equation of given hyperbola with the standard equation
$\dfrac{(y-k)^2}{a^2}-\dfrac{(x-h)^2}{b^2}=1$
$h=1,k=-2,a^2=16,b^2=3$

$e=\sqrt{1+\dfrac{b^2}{a^2}}=\dfrac{\sqrt{19}}{4}$

Distance between the directrices $=\dfrac{2a}{e}=\dfrac{32}{\sqrt{19}}$

Multiple choice mathematics and statistics ellipse standard equation of ellipse introduction to ellipse standard equation of an ellipse

Distance between the foci of the curve represented by the equation $x=3+4\cos\theta, y=2+3\sin\theta$, is?

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

The parametric equations represent an ellipse with semi-major axis a = 4 and semi-minor axis b = 3 centered at (3,2). The distance between the foci of an ellipse is 2ae, where b^2 = a^2(1 - e^2), which simplifies to 2*sqrt(a^2 - b^2) = 2*sqrt(16 - 9) = 2*sqrt(7).

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
D Correct answer
Explanation

To find the point on the ellipse x^2 + 2y^2 = 6 closest to the line x + y = 7, the tangent to the ellipse must be parallel to the line x + y = 7 (slope = -1). The slope of the tangent to the ellipse is given by dy/dx = -x/(2y) = -1, which implies x = 2y. Substituting this into the ellipse equation gives (2y)^2 + 2y^2 = 6, leading to 6y^2 = 6, so y = 1 and x = 2. Thus, the coordinates are (2, 1).

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

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.