Answer to Question #200470 in Analytic Geometry for tanya

Question

Which of the following statements are true and which are false? Give reasons for your answer.

i) 4x ² −9y ² +z ² +36 = 0 represents a hyperboloid of one sheet

ii) The intersection of any plane with an ellipsoid is an ellipse

iii) No plane passes through the points (1,2,3),(1,−1,0) and (1,1,2).

iv) The circle with centre (a,0) and radius a, where a > 0, touches all the sides of the square

x = 0, x = a, y = ±a

v) If the projection of a line segment AB on a line L is 0, then AB lies in L.

Answer 1

i) false

reason:-

4x²− 9y² + z²+ 36 = 0

or

−4x² + 9y²− z² = 36

or

$− \frac{x^2}{3^2} + \frac{y^2}{2^2} − \frac{z^2}{6^2} = 1,$

a hyperboloid of two sheets with axis

the y-axis.

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ii)true

The intersection of a plane and a sphere is a circle (or is reduced to a single point, or is empty). Any ellipsoid is the image of the unit sphere under some affine transformation, and any plane is the image of some other plane under the same transformation. So, because affine transformations map circles to ellipses, the intersection of a plane with an ellipsoid is an ellipse or a single point, or is empty.^[6] Obviously, spheroids contain circles. This is also true, but less obvious, for triaxial ellipsoids (see Circular section). (https://en.wikipedia.org/wiki/Ellipsoid)

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iii)true

A(1;2;3) B(1;-1;0) C(1;1;2)

For drawing up the equation of the plane we use a formula:

$$\begin{vmatrix} x - x_A&y - y_A&z - z_A\\x_B - x_A&y_B - y_A&z_B - z_A\\x_C - x_A&y_C - y_A&z_C - z_A \end{vmatrix} = 0$$

Let's substitute data and we will simplify expression:

$$\begin{vmatrix} x - 1&y - 2&z - 3\\0&-3&-3\\0&-1&-1 \end{vmatrix} = 0$$

(x - 1)(-3·(-1)-(-3)·(-1)) - (y - 2)(0·(-1)-(-3)·0) + (z - 3)(0·(-1)-(-3)·0) = 0

0(x - 1) + 0(y - 2) + 0(z - 3) = 0

As the vector of a normal of the plane is equal to zero, on the given points it is impossible to construct the plane equation.

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iv)false

as the straight line x=a will cross a circle diametrically in two points (a,a) (a,-a)

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v)false

$$|proj_ ba| =\frac{ (a · b)}{|b|}=0$$

the dot product of a projection means and the most direct it is equal to zero, means they are perpendicular