Question

State the type of the quadratic surface: x2+(y6)2+z2=1 1. Hyperboloid of two sheets 2. Hyperboloid of one sheet 3. Ellipsoid 4. None of these

Answer 1

The given quadratic surface equation represents an ellipsoid, as all terms are squared with positive coefficients. This makes the correct answer 'Ellipsoid', which is option 3.

Step-by-step explanation:

The equation given in the question, x2 + (y/6)2 + z2 = 1, represents a quadratic surface. In this equation, the terms for x, y, and z are all squared and have positive coefficients, which means this equation is of the standard form for an ellipsoid. An ellipsoid is a surface where for all points on the surface, the sum of the distances from the point to two fixed points (the foci) is constant. Because the coefficients of x2, (y/6)2, and z2 are positive and the equation is set equal to 1, it is indeed an ellipsoid. The options presented were 'Hyperboloid of two sheets', 'Hyperboloid of one sheet', 'Ellipsoid', and 'None of these'. Hence, the correct answer is option 3, Ellipsoid.

Answer 2

quadratic surface,

x^{2}+z^{2}+\frac{y^{2}}{36}=1

=x^{2}+z^{2}=1-\frac{y^{2}}{36}

=\frac{x^{2}}{a^{2}}+\frac{z^{2}}{c^{2}}=1-\frac{y^{2}}{b^{2}}

this is a hyperbolid of one that intersection at

y=0

the quadratic surface becomes

x^{2}+z^{2}=1

this is the equation of the unit circle

so,the surface is three circle