Answer to Question #92437 in Analytic Geometry for Raghav

Question

Which of the following statements are true and which are false? Give reasons for your
answer.
i) The equation r = acos(θ +α) +bsin(θ +α) represents a circle.
ii) The direction ratios of the line x−5 = 5−y,z = 5 are 1,1,5.
iii) The intersection of a plane and a cone can be a pair of lines.
iv) The angle between the planes x+2y+2z = 5 and 2x+2y+3 = 0 is 60°.
v) The equations 2x^2 +y^2 +3z^2 + 4x + 4y + 18z + 34 = 0, 2x^2 − y^2 = 4y − 4y − 4x represent a real conic.
vi) 4x^2 − 9y^2 + z^2 + 36 = 0 represents a hyperboloid of one sheet.
vii) The intersection of any plane with an ellipsoid is an ellipse.
viii) No plane passes through the points (1,2,3),(1,−1,0) and (1,1,2).
ix) 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.
x) If the projection of a line segment AB on a line L is 0, then AB lies in L.

Answer 1

Solution:

I) true

acos(θ +α) +bsin(θ +α)-> 2ksin(t)

II) false 1,-1,5

III) true

Let's consider the following cone that rests on the  XY plane and whose axis coincides with the Z axis.

Let r be the radius of the base and h be its height.

Without loss of generality, we consider only the planes that are perpendicular to the YZ plane. Two cases are possible:

Case 2: The plane is parallel to the Z

Z axis:

The equation of the plane is

y=c

Then, the points on the intersection of the sides of the cone and the plane are:

$$(h−z)^2/h^2=(x^2+c^2)/r^2$$

Upon rearranging this, we get the form:

$$α(z−h)^2−βx^2=ρ^2$$

which is the equation of a hyperbola (provided c≠0)

When c=0 (that is to say, the plane passes through the apex of the cone), however, we have the form:

$$h−z=±hx/r$$

(https://www.quora.com/What-is-intersection-of-a-true-cone-and-a-plane)

IV) false

Angle between two planes formulas

If A₁x + B₁y + C₁z + D₁ = 0 and A₂x + B₂y + C₂z + D₂ = 0 are a plane equations, then angle between planes can be found using the following formula

$$cos α = \frac{|A_1·A_2 + B_1·B_2 + C_1·C_2|}{\sqrt{A_1^2 + B_1^2 + C_1^2}\sqrt{A_2^2 + B_2^2 + C_2^2}}$$

(https://onlinemschool.com/math/library/analytic_geometry/plane_angl/)

x+2y+2z-5=0 and 2x+2y+3 = 0

$$cos α = \frac{|1·2+ 2·2 + 2·0|}{\sqrt{1^2 + 2^2 + 2^2}\sqrt{2^2 + 2^2 + 0^2}}$$

cos(a)=6/(2√2)=(3√2)/4 -> The angleisn't equal to 60

V) false

2x^2 +y^2 +3z^2 + 4x + 4y + 18z + 34 = 0->2(x+1)^2+(y+2)^2+3(z+3)^2=-1 -imaginary ellipsoid

2x^2 − y^2 = 4y − 4y − 4x -> 2x^2-2y^2+4x=0->2(x+1)^2=y^2+2 - hyperbolic cylinder

VI)false

hyperboloid has the equation

$${\displaystyle {x^{2} \over a^{2}}+{y^{2} \over b^{2}}-{z^{2} \over c^{2}}=1}$$

(https://en.wikipedia.org/wiki/Hyperboloid)

4x^2 − 9y^2 + z^2 + 36 = 0-> x^2/9 + z^2/36− y^2/4 = -1

VII) 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)

VIII) 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.

IX) false

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

X) false

$$|proj_ ba| = (a · b)/|b|=0$$

(https://onlinemschool.com/math/library/vector/projection/)

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