Equation of plane (or maybe its a line ) Linear Algebra

[fattycakez]

The question word for word :

"Write the equation satisfied by all of the points P(x, y, z) that are at the same distance from the point F(0, 0, 4) and the plane z = 0."

I figured I could maybe start by finding the distance between point F and the plane z = 0 but I can't figure out how to represent z=0 as an ax + by + cz + d = 0 equation

I vaguely understand the underlying concepts but I can't quite figure out what the question is asking me to do or how to go about it. I took this question into my community college's tutoring department and even they couldn't figure it out, so any help is greatly appreciated :)

 

[LCKurtz]

The question word for word :

"Write the equation satisfied by all of the points P(x, y, z) that are at the same distance from the point F(0, 0, 4) and the plane z = 0."

I figured I could maybe start by finding the distance between point F and the plane z = 0

What do you need that for? Draw a picture, it's trivial.

but I can't figure out how to represent z=0 as an ax + by + cz + d = 0 equation

Try a = b= d = 0 and c = ?

I vaguely understand the underlying concepts but I can't quite figure out what the question is asking me to do or how to go about it. I took this question into my community college's tutoring department and even they couldn't figure it out, so any help is greatly appreciated :)

Do you understand distance formulas? Set the required two distances equal.

 

[Dick]

The question word for word :

"Write the equation satisfied by all of the points P(x, y, z) that are at the same distance from the point F(0, 0, 4) and the plane z = 0."

I figured I could maybe start by finding the distance between point F and the plane z = 0 but I can't figure out how to represent z=0 as an ax + by + cz + d = 0 equation

I vaguely understand the underlying concepts but I can't quite figure out what the question is asking me to do or how to go about it. I took this question into my community college's tutoring department and even they couldn't figure it out, so any help is greatly appreciated :)

z=0 does have the form ax + by + cz + d = 0. It's a=0, b=0, c=1 and d=0. But you really don't even need that. z=0 is the x-y plane. How far is (x,y,z) from the x-y plane? Just visualize it. Then how far is (x,y,z) from (0,0,4)? Equate the two.

 

[fattycakez]

z=0 does have the form ax + by + cz + d = 0. It's a=0, b=0, c=1 and d=0. But you really don't even need that. z=0 is the x-y plane. How far is (x,y,z) from the x-y plane? Just visualize it. Then how far is (x,y,z) from (0,0,4)? Equate the two.

(x,y,z) is z distance from the x-y plane? And the distance from (x,y,z) to (0,0,4) is sqrt(x^2 + y^2 + (4 - z)^2)?

 

[fattycakez]

I equated the two and got : x^2 + y^2 - 8z + 16 = 0
Does that look correct?
Thanks for the fast replies :)

 

[LCKurtz]

I equated the two and got : x^2 + y^2 - 8z + 16 = 0

That is correct. Solve it for z and see if you recognize it. What kind of surface is it? Then take it to your community college tutors and ask them to explain why they couldn't help you with it.

 

[fattycakez]

Sweet, it looks like a fat 3-d parabola!
Thank you both for your help, I'll let my cc know that strangers on the internet were faster and infinitely more helpful than their tutoring center.

 

[LCKurtz]

Yep. It's called a paraboloid. If you think back you may remember that in 2d a parabola was defined as the locus of points equidistant from the focus (a point) and the directrix (a line). Same idea.