Question

If $A\equiv (0,0),B\equiv (3,4)$ and $AP-BP=5$, then locus of $P$ is

• A. a line
• B. a segment
• C. a pair of ray
• D. none of these

Answer 1

The correct option is A a line

Given : $A\equiv (0,0),B\equiv (3,4)$

$AP-BP=5$

$⟹AP=5+BP$

$⟹A{P}^2=B{P}^2+10BP+25$ ......... Taking square on both sides

$⟹(0-x{)}^2+(0-y{)}^2=(3-x{)}^2+(4-y{)}^2+10BP+25$

$⟹x^2+y^2=9+x^2-6x+16+y^2-8y+10BP+25$

$⟹10BP=6x+8y-50$

$⟹5BP=3x+4y-25$ ............ Dividing both sides by $2$

$⟹25B{P}^2=(3x+4y-25)(3x+4y-25)$ ......... Squaring both the sides

$⟹25\left\{\right(3-x{)}^2+(4-y{)}^2\}=9x^2+12xy-75x+12xy+16y^2-100y-75x-100y+625$

$⟹25(9+x^2-6x+16+y^2-8y)=9x^2+24xy-150x-200y+16y^2+625$

$⟹225+25x^2-150x+400+25y^2-200y=9x^2+24xy-150x-200y+16y^2+625$

$⟹16x^2+9y^2-24xy=0$

$⟹(4x-3y{)}^2=0$

$⟹4x-3y=0$

$\therefore$ Locus of $P$ is $4x-3y=0$ which represents a line.

Hence, option A is correct.