Question

Find the equation of a point which moves so that the sum of its distance from (-4,0) and (4,0) is 10.

I have also attached the answer so explain the steps in a clear way..

Answer 1

Let point P (x,y), A(-4,0) and B(4,0) . Now, it is given in question that sum of AP + AB = 10
Now , Apply the distance formula in place of AP and AB .

Answer 2

Hi ,

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We know that ,

The distance between two points

A( x1 , y1 ) and B( x2 , y2 ) is

AB = √( x2 - x1 )² + ( y2 - y1 )²

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i ) F1( -4 , 0 ) , P( x1 , y1 )

F1P = √ ( x1 + 4 )² + ( y1 - 0 )²---( 1 )

ii ) F2 ( 4 , 0 ) , P ( x1 , y1 )

F2P = √ ( x1 - 4 )² + ( y1 - 0 )² ---( 2 )

It is given that ,

F1P + F2P = 10

√( x1 + 4 )² +y1² + √(x1-4)² + y1² = 10

[(√x1 + 4)² + y1²]² = [10-√(x1-4)²+y1²]²

(x1+4)²+y1²=100 +(x1-4)²+y1²-20√(x1-4)²+y1²

(x1+4)²-(x1-4)² -100 = -20√(x1-4)²+y1²

4x1*4 - 100 = -20√(x1-4)²+y1²

16x1 -100 = -20√(x1-4)²+y1²

divide each term with 4 , we get

4x1 - 25 = -5√(x1-4)²+y1²

do the square both sides of the equation

( 4x1 - 25 )² = 25[ ( x1 - 4 )² + y1² ]

16x1² + 625 -200x1 = 25(x1²+16-8x1+y1² )

16x1² + 625-200x1 =25x1²+400-200x1+25y1²

0 = 9x1² + 25y1² -225

9x1² + 25y1² = 225

divide each term with 225 ,we get

9x1²/225 + 25y1²/225 = 225/225

x1²/16 + y1²/16 = 1

I hope this helps you.

: )