Identifying the equation of a parabola

[Krushnaraj Pandya]

Homework Statement

Find the value of z for which (10x-5)^2 + (10y-7)^2 = z^2((5x+12y+7)^2 is a parabola

Homework Equations

eccentricity of parabola=1

The Attempt at a Solution

I can solve this by expanding everything and writing h^2-ab=0 but this equation looks suspiciously similar to distance from a point=e^2(distance from a line) which should be one for a parabola. Dividing by 10^2 on both sides to remove coefficients of x and y seems reasonable. But I'm not sure how exactly to proceed to identify z easily. I'd appreciate some help, thank you

 

[RPinPA]

I think you're on a good track. So when you divide by $10^2$, on the left you have the expression for $D^2$ where D is the distance from general point (x, y) to some particular point which might be the focus of the parabola.

On the right then you want that expression to be the distance of the same general (x, y) from some line ax + by + c = 0. So you're going to need an expression for the distance of a general point from a general line. Then compare it to your right-hand side. There are a couple of ways to calculate that, from calculus or geometry. Either way it's going to be a lot of algebra. But I'd start there.

There's a general expression for the distance of a point from a line here.
https://en.wikipedia.org/wiki/Distance_from_a_point_to_a_line
That does indeed look like something whose square you could manipulate into looking like your right-hand side, giving you the equation of the directrix and the value of z.

But if this is a homework problem and you don't have that formula in your textbook, you're probably expected to derive it.

 

[Ray Vickson]

Homework Statement

Find the value of z for which (10x-5)^2 + (10y-7)^2 = z^2((5x+12y+7)^2 is a parabola

Homework Equations

eccentricity of parabola=1

The Attempt at a Solution

I can solve this by expanding everything and writing h^2-ab=0 but this equation looks suspiciously similar to distance from a point=e^2(distance from a line) which should be one for a parabola. Dividing by 10^2 on both sides to remove coefficients of x and y seems reasonable. But I'm not sure how exactly to proceed to identify z easily. I'd appreciate some help, thank you

If we write $p$ instead of $z^2$, your equation becomes
$$(10 x -5)^2 +(10 y -7)^2 - p (5x + 12 y + 7)^2 = 0 \hspace{4ex}(1)$$
If you expand out the left-hand-side of (1), you need to determine what values of $p$ give an expression either of the form $A x^2 +Bx + Cy + Dxy + E$ (no $y^2$ term) or $Ay^2 + By + Cx +Dxy + E$ (no $x^2$ term).