Question

find the equation of the locus of a point the difference of whose distances from [-5,0] and [5,0] is 8.​

Answer 1

Let the point be (x,y),

By using distance formula for coordinates,

[(x+5)

2

+(y−0)

2

]

2

1

−[(x−5)

2

+(y−0)

2

]

2

1

=8

⇒[x

2

+10x+25+y

2

]

2

1

=[x

2

−10x+25+y

2

]

2

1

+8

squaring on both sides,

⇒x

2

+10x+25+y

2

=x

2

−10x+25+y

2

+64+16[x

2

−10x+25+y

2

]

2

1

⇒5x−16=4[x

2

−10x+25+y

2

]

2

1

Again squaring both sides,

⇒25x

2

+256−160x=16x

2

−160x+400+16y

2

Therefore, locus is,

9x

2

−16y

2

=144.......(hyperbola)