What is the Equation of the Locus of Points in the Hyperbola or Ellipse Problem?

[Appleton]

1. Homework Statement

The following question is posed within a section of my A level maths book titled "The Hyperbola"

A set of points is such that each point is three times as far from the y-axis as it is from the point (4,0). Find the equation of the locus of P and sketch the locus

2. Homework Equations 3. The Attempt at a Solution

If P is a point (x,y) on the locus, and N is the intersection on the y-axis of the line through P, parallel to the x axis, and S is the point (4,0) then
$PN = 3PS\\ PN^2 = 9PS^2\\ x^2 = 9((x-4)^2+y^2)\\ \frac{8}{9}x^2+y^2-8x+16=0\\$
Which I believe is an elipse, but my book indicates that it is a hyperbola with it's answer of
$8x^2-y^2+8x-16=0$

Is my book wrong?

 

[Ray Vickson]

1. Homework Statement

The following question is posed within a section of my A level maths book titled "The Hyperbola"

A set of points is such that each point is three times as far from the y-axis as it is from the point (4,0). Find the equation of the locus of P and sketch the locus

2. Homework Equations 3. The Attempt at a Solution

If P is a point (x,y) on the locus, and N is the intersection on the y-axis of the line through P, parallel to the x axis, and S is the point (4,0) then
$PN = 3PS\\ PN^2 = 9PS^2\\ x^2 = 9((x-4)^2+y^2)\\ \frac{8}{9}x^2+y^2-8x+16=0\\$
Which I believe is an elipse, but my book indicates that it is a hyperbola with it's answer of
$8x^2-y^2+8x-16=0$

Is my book wrong?

Yes, YOU are right. You can even plot the curve in some package such as Maple to see what is happening.

You can even argue intuitively that the curve must be bounded in the plane, because if you could take $x \to \infty$ very large (and $y$ moderate) on the curve you would have have (approximately) $x \approx 3 (x-4)$, so $x \approx 6$, contradicting the condition that $x \to \infty$ is very large.

 

[Appleton]

Yes, YOU are right. You can even plot the curve in some package such as Maple to see what is happening.

You can even argue intuitively that the curve must be bounded in the plane, because if you could take $x \to \infty$ very large (and $y$ moderate) on the curve you would have have (approximately) $x \approx 3 (x-4)$, so $x \approx 6$, contradicting the condition that $x \to \infty$ is very large.

Thanks, it's encouraging to know that writers of maths books fall prey to the same kind of mistakes that I do, if a little less frequently.

 

[Samy_A]

Thanks, it's encouraging to know that writers of maths books fall prey to the same kind of mistakes that I do, if a little less frequently.

Can happen. The hyperbola they give as result is the solution of the exercise "A set of points is such that each point is three times as far from the y axis point (4,0) as it is from the point (4,0) y axis."

 

[Ray Vickson]

Thanks, it's encouraging to know that writers of maths books fall prey to the same kind of mistakes that I do, if a little less frequently.

It would have been a hyperbola if it had said "... is 3 times as far from the point (4,0) as from the y axis".