find tha coordinates of point which are equidistant from these two pionts p(3.0)and q(-3,0).how many points are possible satisfying the conditon?

Answer :

We can find Co - ordinates of a point that is equidistance from  P ( 3 , 0 )  and Q ( - 3 , 0 ) , As  :

First we draw these two points of x  -  y  plane .

Now , we can see that Every point on y -  axis , is equidistance from given two points P ( 3 , 0 )  and Q ( - 3 , 0 ) .

As  :

We have shown sum points , As  :



O ( 0 , 0 )  , That is 3 unit away from both given points P ( 3 , 0 )  and Q ( - 3 , 0 )  .

C ( 0  , 1 ) , To find distance from both points , we apply distance formula  , As we know 

Distance formula d  = $\sqrt{{\left( x_{2 } - x_{ 1 }\right)}^2 + {\left( y_{2 } - y_{ 1 }\right)}^2}$
So ,
QC  = $\sqrt{{\left( 0 - \left( - 3 \right)\right)}^2 + {\left( 1 -0 \right)}^2} = \sqrt{{\left( 0 + 3 \right)}^2 + {\left( 1 \right)}^2} = \sqrt{9 +1} = \sqrt{10}$ unit 
And
PC = $\sqrt{{\left( 0 - \left( 3 \right)\right)}^2 + {\left( 1 -0 \right)}^2} = \sqrt{{\left( - 3 \right)}^2 + {\left( 1 \right)}^2} = \sqrt{9 +1} = \sqrt{10}$ unit 

So, Point C ( 0 ,1 )  is equidistance from P ( 3 , 0 )  and Q ( - 3 , 0 )  , As we have shown QC =  PC  = $\sqrt{10}$ unit  .

Now we have point D ( 0 , - 1 ) To find distance from both points , we apply distance formula  , As we know 

Distance formula d  = $\sqrt{{\left( x_{2 } - x_{ 1 }\right)}^2 + {\left( y_{2 } - y_{ 1 }\right)}^2}$
So ,
QD  = $\sqrt{{\left( 0 - \left( - 3 \right)\right)}^2 + {\left( -1 -0 \right)}^2} = \sqrt{{\left( 0 + 3 \right)}^2 + {\left( -1 \right)}^2} = \sqrt{9 +1} = \sqrt{10}$ unit 
And
PD = $\sqrt{{\left( 0 - \left( 3 \right)\right)}^2 + {\left( -1 -0 \right)}^2} = \sqrt{{\left( - 3 \right)}^2 + {\left( -1 \right)}^2} = \sqrt{9 +1} = \sqrt{10}$ unit 

So, Point D ( 0 ,- 1 )  is equidistance from P ( 3 , 0 )  and Q ( - 3 , 0 )  , As we have shown QD =  PD  = $\sqrt{10}$ unit  .

Similarly we can show for any point that is lies on y axis is equidistance from P ( 3 , 0 )  and Q ( - 3 , 0 ) .

So, we can say there are infinite number of co - ordinates that are equidistance from P ( 3 , 0 )  and Q ( - 3 , 0 ) .