Q1. Sketch and describe the following surfaces. $$ \begin{array}{llll} ext { (a) } x+y+2 z=4 & ext { (b) } 4 x^{2}+y^{2}=16 & ext { (c) } 4 x^{2}+y^{2}-z^{2}=1 & ext { (d) } z=x\end{array} $$

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Q1. Sketch and describe the following surfaces. $$ \begin{array}{llll}	ext { (a) } x+y+2 z=4 & 	ext { (b) } 4 x^{2}+y^{2}=16 & 	ext { (c) } 4 x^{2}+y^{2}-z^{2}=1 & 	ext { (d) } z=x\end{array} $$

UpStudy AI · Accepted Answer

Let's analyze and describe each of the given surfaces one by one.

### (a) $$ x + y + 2z = 4 $$

**Description:**
This equation represents a plane in three-dimensional space. To understand its orientation, we can rewrite it in the form $$ z = \frac{4 - x - y}{2} $$.

**Sketch:**
- The intercepts can be found by setting two variables to zero:
  - If $$ x = 0 $$ and $$ y = 0 $$, then $$ z = 2 $$ (point (0, 0, 2)).
  - If $$ y = 0 $$ and $$ z = 0 $$, then $$ x = 4 $$ (point (4, 0, 0)).
  - If $$ x = 0 $$ and $$ z = 0 $$, then $$ y = 4 $$ (point (0, 4, 0)).
- The plane will pass through these points and extend infinitely in all directions.

### (b) $$ 4x^2 + y^2 = 16 $$

**Description:**
This equation describes an ellipse in the $$xy$$-plane. To see this more clearly, we can rewrite it as $$ \frac{x^2}{4} + \frac{y^2}{16} = 1 $$, which is the standard form of an ellipse.

**Sketch:**
- The semi-major axis is along the $$y$$-axis with length 4 (from $$y = -4$$ to $$y = 4$$).
- The semi-minor axis is along the $$x$$-axis with length 2 (from $$x = -2$$ to $$x = 2$$).
- The ellipse is centered at the origin (0, 0) in the $$xy$$-plane and extends infinitely in the $$z$$-direction, forming an elliptical cylinder.

### (c) $$ 4x^2 + y^2 - z^2 = 1 $$

**Description:**
This equation represents a hyperboloid of one sheet. It can be rewritten as $$ \frac{4x^2}{1} + \frac{y^2}{1} - \frac{z^2}{1} = 1 $$.

**Sketch:**
- The hyperboloid opens along the $$z$$-axis.
- The cross-sections for constant $$z$$ values will be ellipses when $$z$$ is small and hyperbolas when $$z$$ is larger.
- The surface is symmetric about the $$z$$-axis and extends infinitely in the $$z$$-direction.

### (d) $$ z = x $$

**Description:**
This equation describes a plane in three-dimensional space where the $$z$$-coordinate is equal to the $$x$$-coordinate.

**Sketch:**
- The plane passes through the origin (0, 0, 0) and extends infinitely.
- For any point on this plane, if you know the $$x$$-coordinate, the $$z$$-coordinate is the same, while $$y$$ can take any value.
- The plane is inclined at a 45-degree angle to the $$xy$$-plane and extends infinitely in the $$y$$-direction.

### Summary of Surfaces:
- **(a)** Plane: $$x + y + 2z = 4$$
- **(b)** Elliptical Cylinder: $$4x^2 + y^2 = 16$$
- **(c)** Hyperboloid of One Sheet: $$4x^2 + y^2 - z^2 = 1$$
- **(d)** Plane: $$z = x$$

These surfaces can be visualized in a 3D coordinate system, with each having distinct geometric properties and orientations.
- (a) Plane: $$x + y + 2z = 4$$ - (b) Elliptical Cylinder: $$4x^2 + y^2 = 16$$ - (c) Hyperboloid of One Sheet: $$4x^2 + y^2 - z^2 = 1$$ - (d) Plane: $$z = x$$

Q1. Sketch and describe the following surfaces.

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Answer

Solution

(a)

(b)

©

(d)

Summary of Surfaces:

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