Question

a(ae 0) b(-ae 0) are two points. find the equation to the locus of p such that pa-pb = 2a

Answer 1

Answer with explanation:

Let the coordinate of Point P= (x,y)

It is given that, ⇒P A - P B =2 a-----(1)

Coordinate of point a and b are =A (a e,0) and B(-a e,0)

Using Distance formula,expressing the equation 1

  $\sqrt{(x-ae)^2+(y-0)^2}}+\sqrt{(x+ae)^2+(y-0)^2}}=2 a\\\\\sqrt{(x-ae)^2+(y-0)^2}}=2 a -\sqrt{(x+ae)^2+(y-0)^2}}\\\\ \text{Squaring Both sides}\\\\x^2+a^2e^2-2 a x e+y^2=4 a^2+x^2+a^2e^2+2 a x e+y^2-4 a\times\sqrt{(x+ae)^2+(y-0)^2}}\\\\4 a xe+4 a^2=4 a\times\sqrt{(x+ae)^2+(y-0)^2}}\\\\xe+a=\sqrt{(x+ae)^2+(y-0)^2}}\\\\ \text{Squaring Both sides}\\\\x^2e^2+a^2+2 x e a=x^2+a^2e^2+2 a x e+y^2\\\\x^2\times (e^2-1)-y^2=a^2(e^2-1)\\\\ \frac{x^2}{a^2}-\frac{y^2}{a^2(e^2-1)}=1$

The above equation represent Hyperbola.