**Problems Based on Properties of Distance in Coordinate Geometry :**

Here we are going to see some example problems based on properties of distance in coordinate geometry.

**Question 1 :**

A (–1, 1), B (1, 3) and C (3, a) are points and if AB = BC, then find ‘a’.

**Solution :**

Distance between two points = √(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}

A (–1, 1) and B (1, 3)

= √(1+1)^{2} + (3-1)^{2}

= √2^{2} + 2^{2}

= √(4+4)

AB = √8

B (1, 3) and C (3, a)

= √(1-3)^{2} + (3-a)^{2}

= √(-2)^{2} + (3-a)^{2}

BC = √4 + (3-a)^{2}

AB = BC

√8 = √4 + (3-a)^{2}

Taking squares on both sides

8 = 4 + (3-a)^{2}

8 = 4 + 9 + a^{2} - 6a

8 = 13 + a^{2} - 6a

a^{2} - 6a + 13 - 8 = 0

a^{2} - 6a + 5 = 0

(a - 1)(a - 5) = 0

a = 1 and a = 5

**Question 2 :**

The abscissa of a point A is equal to its ordinate, and its distance from the point B(1, 3) is 10 units, What are the coordinates of A?

**Solution :**

Since the point A is equal to its ordinates

x = y

Let A(x, x)

Distance between the points A and B :

A(x, x) B(1, 3)

√(1-x)^{2} + (3-x)^{2 } = 10

Taking squares on both sides

(1-x)^{2} + (3-x)^{2 } = 100

1 - 2x + x^{2} + 9 - 6x + x^{2} = 100

2x^{2} - 8x + 10 - 100 = 0

2x^{2} - 8x - 90 = 0

Divide the entire equation by 2, we get

x^{2} - 4x - 45 = 0

(x - 9)(x + 5) = 0

x = 9 and x = -5

A(9, 9) or B(-5, -5)

**Question 3 :**

The point (x, y) is equidistant from the points (3, 4) and

(–5,6). Find a relation between x and y.

**Solution :**

Let the points be p (x, y) A(3, 4) and B(-5, 6)

PA = PB

√(3-x)^{2} + (4-y)^{2 } = √(-5-x)^{2} + (6-y)^{2 }

Taking squares on both sides

(3-x)^{2} + (4-y)^{2 } = (5+x)^{2} + (6-y)^{2 }

9 - 6x + x^{2} + 16 - 8y + y^{2} = 25 + 10x + x^{2} + 36 - 12y + y^{2}

-6x - 10x - 8y + 12y + 25 - 61 = 0

-16x + 4y - 36 = 0

4x - y + 9 = 0

y = 4x + 9

**Question 4 :**

Let A(2, 3) and B(2, –4) be two points. If P lies on the x-axis, such that AP = (3/7)AB, find the coordinates of P.

**Solution :**

Since the point P lies on x-axis, the y-coordinate will be 0.

A(2, 3) P(x, 0)

= √(x - 2)^{2} + (0 - 3)^{2 }

= √(x - 2)^{2} + 9 ------(1)

A(2, 3) B(2, -4)

= √(2 - 2)^{2} + (-4 - 3)^{2 }

= √0^{2} + (-7)^{2}

= 7 ------(2)

AP = (3/7)AB

√(x - 2)^{2} + 9 = (3/7) 7

(x - 2)^{2} + 9 = 3^{2}

x^{2} - 4x + 4 + 9 - 9 = 0

x^{2} - 4x + 4 = 0

(x - 2) (x - 2) = 0

x = 2 and x = 2

Hence the point P is (2, 0)

**Question 5 :**

Show that the point (11, 2) is the centre of the circle passing through the points (1, 2), (3, –4) and (5, –6)

**Solution :**

Let the center point be O (11, 2)

A(1, 2) B(3, -4) and C(5, -6)

AO = BO = CO

O (11, 2) and A(1, 2)

= √(1 - 11)^{2} + (1 - 11)^{2}

= √100 + 100

= √200

O (11, 2) and B(3, -4)

= √(3 - 11)^{2} + (-4 - 2)^{2}

= √(-8)^{2} + (-6)^{2}

= √64 + 36

= √100

O (11, 2) and C (5, –6)

= √(5 - 11)^{2} + (-6 - 2)^{2}

= √(-6)^{2} + (-8)^{2}

= √36 + 64

= √100

Hence the given point (11, 2) is the center of the circle.

**Question 6 :**

The radius of a circle with centre at origin is 30 units. Write the coordinates of the points where the circle intersects the axes. Find the distance between any two such points.

**Solution :**

From the diagram given above, we come to know that the distance between the points (0, 0) and (a, 0) is 30 units.

√(0 - a)^{2} + (0 - 0)^{2 } = 30

√(-a)^{2} = 30

a = 30

it is clearly shown that circle intersects the co-ordinate axes at four points. and that are (30,0) , (-30,0) ,(0,30) and (0,-30).

now, distance between (30,0) and (0,30)

= √(30-0)² + (0 - 30)² = 30√2 unit

Similarly , you can find distance between any such two points.

for better understanding, let A = (30,0) , B=(0,30) , C = (-30,0) and D = (0, -30)

then, Length of AB = length of BC = length of CD = length of DA = 30√2 unit

After having gone through the stuff given above, we hope that the students would have understood, "Problems Based on Properties of Distance in Coordinate Geometry"

Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.

Widget is loading comments...

You can also visit our following web pages on different stuff in math.

**WORD PROBLEMS**

**Word problems on simple equations **

**Word problems on linear equations **

**Word problems on quadratic equations**

**Area and perimeter word problems**

**Word problems on direct variation and inverse variation **

**Word problems on comparing rates**

**Converting customary units word problems **

**Converting metric units word problems**

**Word problems on simple interest**

**Word problems on compound interest**

**Word problems on types of angles **

**Complementary and supplementary angles word problems**

**Markup and markdown word problems **

**Word problems on mixed fractrions**

**One step equation word problems**

**Linear inequalities word problems**

**Ratio and proportion word problems**

**Word problems on sets and venn diagrams**

**Pythagorean theorem word problems**

**Percent of a number word problems**

**Word problems on constant speed**

**Word problems on average speed **

**Word problems on sum of the angles of a triangle is 180 degree**

**OTHER TOPICS **

**Time, speed and distance shortcuts**

**Ratio and proportion shortcuts**

**Domain and range of rational functions**

**Domain and range of rational functions with holes**

**Graphing rational functions with holes**

**Converting repeating decimals in to fractions**

**Decimal representation of rational numbers**

**Finding square root using long division**

**L.C.M method to solve time and work problems**

**Translating the word problems in to algebraic expressions**

**Remainder when 2 power 256 is divided by 17**

**Remainder when 17 power 23 is divided by 16**

**Sum of all three digit numbers divisible by 6**

**Sum of all three digit numbers divisible by 7**

**Sum of all three digit numbers divisible by 8**

**Sum of all three digit numbers formed using 1, 3, 4**

**Sum of all three four digit numbers formed with non zero digits**