CameraIcon
CameraIcon
SearchIcon
MyQuestionIcon
MyQuestionIcon
Question

Find the equation of the set of points P, the sum of whose distances from A (4, 0, 0) and B (–4, 0, 0) is equal to 10.

Open in App
Solution

The given points are A=( 4,0,0 ) and B=( 4,0,0 ) .

Let P=( x,y,z ) be a point the sum of whose distances from A and B is 10units .

The formula to find the distance d between two points ( x 1 , y 1 , z 1 ) and ( x 2 , y 2 , z 2 ) is,

d= ( x 2 x 1 ) 2 + ( y 2 y 1 ) 2 + ( z 2 z 1 ) 2 

To find the distance AP between the points A and P, the value of x 1 is 4 , x 2 is x , y 1 is 0 , y 2 is y , z 1 is 0 , z 2 is z .

Substitute the value of x 1 , x 2 , y 1 , y 2 , z 1 and z 2 in equation (1), to find the distance AP

AP= ( x4 ) 2 + y 2 + z 2

To find the distance BP between the points B and P, the value of x 1 is 4 , x 2 is x , y 1 is 0 , y 2 is y , z 1 is 0 , z 2 is z .

Substitute the value of x 1 , x 2 , y 1 , y 2 , z 1 and z 2 in equation (1), to find the distance BP

AP= ( x+4 ) 2 + y 2 + z 2

It is given that, AP+BP=10 . Substitute values,

( x4 ) 2 + y 2 + z 2 + ( x+4 ) 2 + y 2 + z 2 =10 ( x4 ) 2 + y 2 + z 2 =10 ( x+4 ) 2 + y 2 + z 2

Squaring both sides,

( ( x4 ) 2 + y 2 + z 2 ) 2 = ( 10 ( x+4 ) 2 + y 2 + z 2 ) 2 ( x4 ) 2 + y 2 + z 2 =100+ ( x+4 ) 2 + y 2 + z 2 20 ( x+4 ) 2 + y 2 + z 2 x 2 x 2 8x8x+1616+ y 2 + z 2 y 2 z 2 =10020 ( x+4 ) 2 + y 2 + z 2 100+16x=20 ( x+4 ) 2 + y 2 + z 2

Squaring both sides again,

( 100+16x ) 2 = ( 20 ( x+4 ) 2 + y 2 + z 2 ) 2 ( 25+4x ) 2 = ( 5 ( x+4 ) 2 + y 2 + z 2 ) 2 625+16 x 2 +200x=25 x 2 +200x+400+25 y 2 +25 z 2 9 x 2 +25 y 2 +25 z 2 225=0

Thus, the equation of the set of points P, the sum of whose distances from the points A=( 4,0,0 ) and B=( 4,0,0 ) is 10units is 9 x 2 +25 y 2 +25 z 2 225=0 .


flag
Suggest Corrections
thumbs-up
1
BNAT
mid-banner-image