# Show that the path traced by the point of intersection of three mutual perpendicular tangent planes to the ellipsoid #ax^2+by^2+cz^2=1# is a sphere with the same centre as that of the ellipsoid.?

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Show that the path traced by the point of intersection of three mutual perpendicular tangent planes to the ellipsoid #ax^2+by^2+cz^2=1# is a sphere with the same centre as that of the ellipsoid.

Show that the path traced by the point of intersection of three mutual perpendicular tangent planes to the ellipsoid

##### 1 Answer

#### Answer:

See below.

#### Explanation:

Calling

If

Let

but

Now given three orthogonal planes

and calling

and as a consequence

then we have also

Now adding

and finally

but

so

which is the path traced by the point of intersection of three mutual perpendicular tangent planes to the ellipsoid.

Attached a plot for the ellipsoid