Coordinates of a point that touches the ellipsoid

[ppmko]

Homework Statement

I have an ellipsoid with center (000). There is a point A inside the ellipsoid with known coordinates(1,2,3) I draw a line from center to point A and extend it to cut the ellipsoid on on point p(x,y,z).




2. Homework Equations

I want to find the coordinates of point P(x,y,z)


3. The Attempt at a Solution


The equation of ellipsoid for p is x^2/a^2+y^2/b^2+z^2/c^2=1
i have the values of a,b and c
i want to know if the ellipsoid equation is applicable to coordinates of A and coordinates of P

and how can i create equation using the coordinates of p with the coordinates of A
by equation of line method as both the points lie on a straight line with one end on (000) as the third point.

 

[jvicens]

Firstly, you can use the equation of the line to find the coordinates of point P. The equation of the line can be written as:

x = x0 + at
y = y0 + bt
z = z0 + ct

Where (x0, y0, z0) is the center of the ellipsoid (000) and (a, b, c) is the direction vector from the center to point A.

Substituting the known values, we get:

x = 0 + 1t = t
y = 0 + 2t = 2t
z = 0 + 3t = 3t

Now, we can substitute these values into the equation of the ellipsoid to find the coordinates of point P.

(t)^2/a^2 + (2t)^2/b^2 + (3t)^2/c^2 = 1

Simplifying, we get:

t^2 (1/a^2 + 4/b^2 + 9/c^2) = 1

Therefore, t = ± 1/√(1/a^2 + 4/b^2 + 9/c^2)

Substituting this value of t back into the equations for x, y, and z, we get the coordinates for point P:

x = ± 1/√(1/a^2 + 4/b^2 + 9/c^2)
y = ± 2/√(1/a^2 + 4/b^2 + 9/c^2)
z = ± 3/√(1/a^2 + 4/b^2 + 9/c^2)

Therefore, the coordinates of point P are: (± 1/√(1/a^2 + 4/b^2 + 9/c^2), ± 2/√(1/a^2 + 4/b^2 + 9/c^2), ± 3/√(1/a^2 + 4/b^2 + 9/c^2)).

You can also use the distance formula to find the coordinates of point P. The distance from the center (000) to point P is equal to the distance from the center (000) to point A.

Therefore, (x - 0)^2 + (y