14 a Find an equation for and identify the surface that resu

14. a) Find an equation for, and identify the surface that results when the elliptic cone 4a 2+ 9y 0 is reflected about the plane: i) a 0, ii) y 0, iii) 2 0, iv) a y, v) y 2, vi) z z. b) Identify each quadric surface: i) 9ac2 4y2 22 1, ii) y r2 22, iii) 2 (r 2)2 4(y 3)2, iv) 10, v) z 2y 42, vi) 4r y 16(2-2) 100

Solution

This question is a heavily based on visualization of the curves.

let us define our curve as C : 4x²+9y²-25z² .

Clearly this is a elliptic cone with boundaries defined by y=+-5/3z ; x=+-5/2z and x=+-3/2y .

Kindly note that the three boundaries lines of the cone pass through the origin (0,0,0) thus the two vertices of the cone (summit) will meet each other at the origin.

Now Imagine the shape of this cone?

This cone will be symmetrical across all three axes and also along all three planes .

Now let us take plane x

The upper part will be a reflection of lower part and vice versa .Thus the equation of the reflection will be nothing but equal to the actual equation of the conic (cone in this case) .

Thus equation of reflection of the elliptic cone against the x,y and z planes will be equal to the initial equation of the cone.

b). Identification of the quadratic curves :-

i. 9x²-4y²-z²=1  

Clearly the above mentioned equation will be that of hyperbola with its curve touching the z axis at unit distance from the origin on both sides., y axis at 1/2 unit and x axis at 1/3 units from origin.

From the equation we can note that the lines defining the boundary are x=+-(1+z²)^1/2/3; y=+-(1+4y²)^1/2/3 and z=+-(9x²+1)^1/2

ii. y=x²-z²

^wWhen y=0 , x=+-z,

when x=0, y=-z² (inverted parabola )

^when z=0, y=x² (parabola)

Thus from the three boundary conditions we get that the resulting shape will be a paraboloid bounded by line x=+-z

iii.z= (x-2)²+4(y+3)² clearly this is also a paraboloid with the same logic as applied in ii.

iv. 9x²+y²+z²-18x+2y+16z=10

this can be represented as 9(x-1)² +(y+1)²+(z+8)²=84

or (x-1)²/(84/9)+(y+1)²/(84)+(z+8)²/(84)=1

clearly this is of the form (x/a)²+(y/b)²+(z/c)²=1 which is a generic equation of an elliptoid.

Thus this is an equation of an elliptiod .

v. z²=4x²+y²+8x-2y+4z

this can be written as

(x+1)²/(1/2)²+(y-1)²-(z-2)² =1

This is an equation of an elliptic cone with its axis of rotation along the z axis as shown in part 1 and i of the problem above.

vi. 4x²-y²+16(z-2)²=100

This can be written as

(x/5)²-(y/10)²+((z-2)/2.5)² =1

This is a cone with its axis of rotation along y axis