The surface 𝑥2+𝑦2−3𝑥+𝑧=4 is symmetric with respect to:Group of answer choicesxy-planexz-planeyz-planey-axis
Question
The surface 𝑥2+𝑦2−3𝑥+𝑧=4 is symmetric with respect to:Group of answer choicesxy-planexz-planeyz-planey-axis
Solution
The given surface equation is 𝑥²+𝑦²−3𝑥+𝑧=4.
To determine the symmetry, we can rewrite the equation in a more standard form.
First, complete the square for the x-terms: (x² - 3x) can be rewritten as (x - 3/2)² - (3/2)².
So, the equation becomes:
(x - 3/2)² + y² + z = 4 + (3/2)²
Simplify the right side:
(x - 3/2)² + y² + z = 4 + 9/4 = 25/4
So, the equation in standard form is:
(x - 3/2)² + y² + z = 25/4
This is the equation of a cylinder that is symmetric with respect to the yz-plane.
So, the answer is yz-plane.
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Find the axis of symmetry
Find a normal vector n𝑛 to the tangent plane to the surfacez=−3x2y3𝑧=−3𝑥2𝑦3 at the point (x,y,z)=(−3,4,−1728)(𝑥,𝑦,𝑧)=(−3,4,−1728) and enter your answer in the box below using Maple syntax. n=𝑛=
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