Question

In what ratio does the centre O (3 , 3) divides the line segment joining the points P and Q?​

Answer 1

Answer:

I guess it is 1:1

Step-by-step explanation:

Answer 2

SOLUTION

COMPLETE QUESTION

In a city, a circular park is situated with centre O(3 , 3). There are two exit gates P and Q which are opposite to each other. The location of exit gate ‘P’ is (5 , 3).

QUESTION 1:

The location of exit gate ‘Q’ will be:-

(a) (3 , 1)

(b) (3 , 3)

(c) (1 , 3)

(d) (5 , 3)

QUESTION 2:

What will be the distance between two exit gates P and Q

(a) 3 units

(b) 4 units

(c) 5 units

(d) 6 units

QUESTION 3:

If a pole R( x , 5) is standing on a boundary of circular park which is equidistant from Pand Q then, the value of ‘x’ will be

(a) 0

(b) 1

(c) 2

(d) 3

QUESTION 4:

In what ratio does the centre O (3 , 3) divides the line segment joining the points P and Q

(a) 1:1

(b) 1:2

(c) 2:1

(d) 1:4

FORMULA TO BE IMPLEMENTED

For the given two points $\sf{A( x_1 , y_1) \: \: and \: \: B( x_2 , y_2)}$

1. The midpoint of the line AB is

$\displaystyle \sf{ \bigg( \frac{x_1 + x_2}{2} , \frac{y_1 + y_2}{2} \bigg)}$

2. The distance between the points

$= \sf{ \sqrt{ {(x_2 -x_1 )}^{2} + {(y_2 -y_1 )}^{2} } }$

EVALUATION

Here it is given that circular park is situated with centre O(3 , 3).

There are two exit gates P and Q which are

opposite to each other.

The location of exit gate P is (5 , 3).

ANSWER TO QUESTION : 1

Since two exit gates P(5,3) and Q are

opposite to each other.

The centre is O(3 , 3)

Let (a, b) be the coordinates of the point Q

So O is the middle point of the line PQ

So by the given condition

$\displaystyle \sf{ \frac{5 + a}{2} = 3 \: \: and \: \: \frac{3 + b}{2} = 3 }$

$\displaystyle \sf{ \implies 5 + a = 6 \: \: and \: \: 3 + b= 6}$

$\displaystyle \sf{ \implies a = 1 \: \: and \: \: b= 3}$

So the coordinates of the point Q is (1,3)

Hence the correct option is (c) (1 , 3)

ANSWER TO QUESTION : 2

The coordinates of the point P is (5,3)

The coordinates of the point Q is (1,3)

The distance between two exit gates P and Q

$\sf = \sqrt{ {(5 - 1)}^{2} + {(3 - 3)}^{2} } \: \: \: unit$

$\sf = \sqrt{ {(4)}^{2} + {(0)}^{2} } \: \: \: unit$

$\sf = 4 \: \: \: unit$

Hence the correct option is (b) 4 units

ANSWER TO QUESTION : 3

The coordinates of the point P is (5,3)

The coordinates of the point Q is (1,3)

The pole R( x , 5) is standing on a boundary of circular park which is equidistant

from P and Q

So by the given condition

RP = RQ

$\displaystyle \sf{ \implies \sqrt{ {(x - 5)}^{2} + {(5 - 3)}^{2} } = \sqrt{ {(x - 1)}^{2} + {(5 - 3)}^{2} } }$

$\displaystyle \sf{ \implies {(x - 5)}^{2} + {(5 - 3)}^{2} = {(x - 1)}^{2} + {(5 - 3)}^{2} }$

$\displaystyle \sf{ \implies {(x - 5)}^{2} + {(2)}^{2} = {(x - 1)}^{2} + {(2)}^{2} }$

$\displaystyle \sf{ \implies {(x - 5)}^{2} = {(x - 1)}^{2} }$

$\displaystyle \sf{ \implies {x}^{2} - 10x + 25 = {x }^{2} - 2x + 1 }$

$\displaystyle \sf{ \implies - 8x = - 24 }$

$\displaystyle \sf{ \implies x = 3 }$

The value of x = 3

Hence the correct option is (d) 3

ANSWER TO QUESTION : 4

Since two exit gates P(5,3) and Q are

opposite to each other.

The centre is O(3 , 3)

So O is the middle point of the line PQ

Thus O divides the line segment joining the points P

and Q in the ratio 1 : 1

Hence the correct option is (a) 1 : 1

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