Question #102908
Reduce the following equations to standard form, and then identify which conicoids they represent. Further, give a rough sketch of each.
i) x^2 +y^2 +2x−y−z+3 = 0
ii) 3y^2 +3z^2 +4x+3y+z = 9
1
Expert's answer
2020-03-19T17:02:14-0400

i) x2+y2+2xyz+3=0x^2+y^2+2x-y-z+3=0

then we group variables. We get equation. (x+1)2+(y1/2)2+7/4=z(x+1)^2+(y-1/2)^2+7/4=z

Finish formul is (x+1)2+(y1/2)2=z7/4(x+1)^2+(y-1/2)^2=z-7/4

This formula is the formula of elliptic paraboloid (we then can see from standard form of elliptic paraboloid, zz0=A(xx0)2+B(yy0)2z-z_0=A(x-x_0)^2+B(y-y_0)^2 , where the coefficients A and B have the same sings )

Picture



ii) 3y2+3z2+4x+3y+z=93y^2 +3z^2 +4x+3y+z = 9

Then, also we do group variables. We get the formula. 3(y+1/2)23/4+3(z+1/6)21/12=94x3(y+1/2)^2-3/4+3(z+1/6)^2-1/12=9-4x3(y+1/2)2+3(z+1/6)2=49/64x3(y+1/2)^2+3(z+1/6)^2=49/6-4x All numbers and variable x on the right side.

This is also elliptic paraboloid, but this elliptic paraboloid going on x-os.

Picture



Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!
LATEST TUTORIALS
APPROVED BY CLIENTS