Answer:
The equation of tangent plane to the hyperboloid
[tex]\frac{xx_0}{a^2}+\frac{yy_0}{b^2}-\frac{zz_0}{c^2}=1[/tex].
Step-by-step explanation:
Given
The equation of ellipsoid
[tex]\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1[/tex]
The equation of tangent plane at the point [tex]\left(x_0,y_0,z_0\right)[/tex]
[tex]\frac{xx_0}{a^2}+\frac{yy_0}{b^2}+\frac{zz_0}{c^2}=1[/tex] ( Given)
The equation of hyperboloid
[tex]\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1[/tex]
F(x,y,z)=[tex]\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}[c^2}[/tex]
[tex] F_x=\frac{2x}{a^2},F_y=\frac{2y}{b^2},F_z=-\frac{2z}{c^2}[/tex]
[tex] (F_x,F_y,F_z)(x_0,y_0,z_0)=\left(\frac{2x_0}{a^2},\frac{2y_0}{b^2},-\frac{2z_0}{c^2}\right)[/tex]
The equation of tangent plane at point [tex]\left(x_0,y_0,z_0\right)[/tex]
[tex]\frac{2x_0}{a^2}(x-x_0)+\frac{2y_0}{b^2}(y-y_0)-\farc{2z_0}{c^2}(z-z_0)=0[/tex]
The equation of tangent plane to the hyperboloid
[tex]\frac{2xx_0}{a^2}+\frac{2yy_0}{b^2}-\frac{2zz_0}{c^2}-2\left(\frac{x_0^2}{a^2}+\frac{y_0^2}{b^2}-\frac{z_0^2}{c^2}\right)=0[/tex]
The equation of tangent plane
[tex]2\left(\frac{xx_0}{a^2}+\frac{yy_0}{b^2}-\frac{zz_0}{c^2}\right)=2[/tex]
Hence, the required equation of tangent plane to the hyperboloid
[tex]\frac{xx_0}{a^2}+\frac{yy_0}{b^2}-\frac{zz_0}{c^2}=0[/tex]
The equation of the tangent plane to the ellipsoid at the given point is [tex]\frac{xx^0}{a^2} + \frac{yy^0}{b^2} - \frac{zz^0}{c^2} = 1[/tex]
The equation of the tangent plane to the ellipsoid is given as:
[tex]\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1[/tex]
The point on the ellipsoid is given as:
(x0, y0, z0)
The equation of the tangent plane to the ellipsoid at the given point can be written as:
[tex]\frac{xx^0}{a^2} + \frac{yy^0}{b^2} + \frac{zz^0}{c^2} = 1[/tex]
Given that the equation of the tangent plane to the hyperboloid is
[tex]\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1[/tex]
The equations at the tangents of the ellipsoid and the hyperboloid take the same form.
So, the equation of the tangent plane to the ellipsoid at the given point is [tex]\frac{xx^0}{a^2} + \frac{yy^0}{b^2} - \frac{zz^0}{c^2} = 1[/tex]
Read more about tangent planes at:
https://brainly.com/question/15465847
