Question

Find the equation of a curve that is four more units away from (-10, 0) than (10,0)

Answer 1

So first you shlould try to transform those information on a formula, the formula for distance is:\[d=\sqrt{(x-x_0)^2+(y-y_0)^2}\]What the problem is saying is that the distance from the curve to (-10, 0) is four units more then the distance to (10, 0) so:\[\sqrt{(x-(-10))^2+y^2}=4+\sqrt{(x-10)^2+y^2}\]Now we only need to change this equation so that y is written as a function of x:\[(x+10)^2+y^2=\left(4+\sqrt{(x-10)^2+y^2}\right)^2=16+8\sqrt{(x-10)^2+y^2}+(x-10)^2+y^2\]\[40x=16+8\sqrt{(x-10)^2+y^2}\]\[\left(\frac{40x-16}{8}\right)^2=(x-10)^2+y^2\]\[(5x-2)^2=25x^2-20x+4=(x-10)^2+y^2=x^2-20x+100+y^2\]\[24x^2-96=y^2\]\[y=\sqrt{24x^2-96}\]

Answer 2

is the curve a hyperbola?

Answer 3

\[\frac{ x^2 }{ 4 } - \frac{ y^2 }{ 96 } = 1\]

Answer 4

That is correct