Problems on Distance Formula

We will discuss here how to solve the problems on distance formula.

The distance between two points A (x1, y1) and B (x2, y2) is given by the formula

AB = (x1x2)2+(y1y2)2


1. If the distance between the points (5, - 2) and (1, a) is 5, find the values of a.

Solution:

We know, the distance between (x1, y1) and (x2, y2)

is (x1x2)2+(y1y2)2

Here, the distance = 5, x1 = 5, x2 = 1, y1 = -2 and y2 = a

Therefore, 5 = (51)2+(2a)2

⟹ 25 = 16 + (2 + a)2

⟹ (2 + a)2 = 25 - 16

⟹ (2 + a)2 = 9

Taking square root, 2 + a = ±3

⟹ a = -2 ± 3

⟹ a = 1, -5


2. The co-ordinates of points on the x-axis which are at a distance of 5 units from the point (6, -3).

Solution:

Let the co-ordinates of the point on the x-axis be (x, 0)

Since, distance = (x2x1)2+(y2y1)2

Now taking (6, -3) = (x1, y1) and (x, 0) = (x2, y2), we get

5 = (x6)2+(0+3)2

Squaring both sides we get

⟹ 25 = (x – 6)2 + 32

⟹ 25 = x2 – 12x + 36 + 9

⟹ 25 = x2 – 12x + 45

⟹ x2 – 12x + 45 – 25 = 0

⟹ x2 – 12x + 20 = 0

⟹ (x – 2)(x – 10) = 0

⟹ x = 2 or x = 10

Therefore, the required points on the x-axis are (2, 0) and (10, 0).


3. Which point on the y-axis is equidistance from the points (12, 3) and (-5, 10)?

Solution:

Let the required point on the y-axis (0, y).

Given (0, y) is equidistance from (12, 3) and (-5, 10)

i.e., distance between (0, y) and (12, 3) = distance between (0, y) and (-5, 10)

(120)2+(3y)2 = (50)2+(10y)2

⟹ 144 + 9 + y2 – 6y = 25 + 100 + y2 – 20y

⟹ 14y = -28

⟹ y = -2

Therefore, the required point on the y-axis = (0, -2)


4. Find the values of a such that PQ = QR, where P, Q and R are the points whose coordinates are (6, - 1), (1, 3) and (a, 8) respectively.

Solution:

PQ = (61)2+(13)2

     = 52+(4)2

     = 25+16

     = 41

QR = (1a)2+(38)2

     = (1a)2+(5)2

     = (1a)2+25

Therefore, PQ = QR

41 = (1a)2+25

⟹ 41 = (1 - a)2 + 25

⟹ (1 - a)2 = 41 - 25

⟹ (1 - a)2 = 16

⟹ 1 - a = ±4

⟹ a = 1 ±4

⟹ a = -3, 5


5. Find the points on the y-axis, each of which is at a distance of 13 units from the point (-5, 7).

Solution:

Let A (-5, 7) be the given point and let P (0, y) be the required point on the y-axis. Then,

PA = 13 units

⟹ PA2 = 169

⟹ (0 + 5)2 + (y - 7)2 = 169

⟹ 25 + y2 - 14y + 49 = 169

⟹ y2 – 14y + 74 = 169

⟹ y2 – 14y – 95 = 0

⟹ (y - 19)(y + 5) = 0

⟹ y – 19 = 0 or, y + 5 = 0

⟹ y = 19 or, y = -5

Hence, the required points are (0, 19) and (0, -5)

 Distance and Section Formulae





10th Grade Math


From Problems on Distance Formula to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.




Share this page: What’s this?

Recent Articles

  1. Multiplication by Ten, Hundred and Thousand |Multiply by 10, 100 &1000

    May 01, 25 11:57 PM

    Multiply by 10
    To multiply a number by 10, 100, or 1000 we need to count the number of zeroes in the multiplier and write the same number of zeroes to the right of the multiplicand. Rules for the multiplication by 1…

    Read More

  2. Adding and Subtracting Large Decimals | Examples | Worksheet | Answers

    May 01, 25 03:01 PM

    Here we will learn adding and subtracting large decimals. We have already learnt how to add and subtract smaller decimals. Now we will consider some examples involving larger decimals.

    Read More

  3. Converting Fractions to Decimals | Solved Examples | Free Worksheet

    Apr 28, 25 01:43 AM

    Converting Fractions to Decimals
    In converting fractions to decimals, we know that decimals are fractions with denominators 10, 100, 1000 etc. In order to convert other fractions into decimals, we follow the following steps:

    Read More

  4. Expanded Form of a Number | Writing Numbers in Expanded Form | Values

    Apr 27, 25 10:13 AM

    Expanded Form of a Number
    We know that the number written as sum of the place-values of its digits is called the expanded form of a number. In expanded form of a number, the number is shown according to the place values of its…

    Read More

  5. Converting Decimals to Fractions | Solved Examples | Free Worksheet

    Apr 26, 25 04:56 PM

    Converting Decimals to Fractions
    In converting decimals to fractions, we know that a decimal can always be converted into a fraction by using the following steps: Step I: Obtain the decimal. Step II: Remove the decimal points from th…

    Read More