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Consider x² y² 100 25 the following equation of a quadric surface. . Find the intercepts with the three coordinate axes, if they exist. . Find the equations of the xy-, xz-, and yz-traces, if they exist. . Identify and sketch a graph of the surface. =z² A. The surface intersects the x-axis at x = 0. (Use a comma to separate answers as needed.) B. There are no x-intercepts. Find the y-intercepts, if they exist. Select the correct choice and fill in any answer boxes within your choice. A. The surface intersects the y-axis at y = 0. (Use a comma to separate answers as needed.) B. There are no y-intercepts. Find the z-intercepts, if they exist. Select the correct choice and fill in any answer boxes within your choice. A. The surface intersects the z-axis at z = 0. (Use a comma to separate answers as needed.) B. There are no z-intercepts. . Find the equation of the xy-trace, if it exists. Select the correct choice and fill in any answer boxes within your choice. XA. The trace is not a single point. The equation of the xy-trace is (Type an equation.) B. The trace is the single point (0,0,0) Oc. There is no xy-trace. Find the equation of the xz-trace, if it exists. Select the correct choice and fill in any answer boxes within your choice. OA. The trace is not a single point. The equation of the xz-trace is (Type an equation.) ● B. The trace is the single point. c. There is no xz-trace.

Consider x² y² 100 25 the following equation of a quadric surface. . Find the intercepts with the three coordinate axes, if they exist. . Find the equations of the xy-, xz-, and yz-traces, if they exist. . Identify and sketch a graph of the surface. =z² A. The surface intersects the x-axis at x = 0. (Use a comma to separate answers as needed.) B. There are no x-intercepts. Find the y-intercepts, if they exist. Select the correct choice and fill in any answer boxes within your choice. A. The surface intersects the y-axis at y = 0. (Use a comma to separate answers as needed.) B. There are no y-intercepts. Find the z-intercepts, if they exist. Select the correct choice and fill in any answer boxes within your choice. A. The surface intersects the z-axis at z = 0. (Use a comma to separate answers as needed.) B. There are no z-intercepts. . Find the equation of the xy-trace, if it exists. Select the correct choice and fill in any answer boxes within your choice. XA. The trace is not a single point. The equation of the xy-trace is (Type an equation.) B. The trace is the single point (0,0,0) Oc. There is no xy-trace. Find the equation of the xz-trace, if it exists. Select the correct choice and fill in any answer boxes within your choice. OA. The trace is not a single point. The equation of the xz-trace is (Type an equation.) ● B. The trace is the single point. c. There is no xz-trace.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Consider the following equation of a quadric surface: \[ \frac{x^2}{100} + \frac{y^2}{25} = z^2 \] **a. Find the intercepts with the three coordinate axes, if they exist.** - **x-intercepts:** - A. The surface intersects the x-axis at \( x = 0 \). - B. There are no x-intercepts. - **y-intercepts:** - A. The surface intersects the y-axis at \( y = 0 \). - B. There are no y-intercepts. - **z-intercepts:** - A. The surface intersects the z-axis at \( z = 0 \). - B. There are no z-intercepts. **b. Find the equations of the xy-, xz-, and yz-traces, if they exist.** - **xy-trace:** - A. The trace is not a single point. The equation of the xy-trace is [blank]. - B. The trace is the single point \( (0, 0, 0) \). - C. There is no xy-trace. - **xz-trace:** - A. The trace is not a single point. The equation of the xz-trace is [blank]. - B. The trace is the single point \( (0, 0, 0) \). - C. There is no xz-trace. There is no information provided for the yz-trace. **c. Identify and sketch a graph of the surface.** The problem does not provide a sketch; however, based on the equation, the surface is indicative of an elliptic cone centered at the origin. This representation helps in understanding the geometrical shape and characteristics of the surface in a 3D coordinate system.
Transcribed Image Text:Consider the following equation of a quadric surface: \[ \frac{x^2}{100} + \frac{y^2}{25} = z^2 \] **a. Find the intercepts with the three coordinate axes, if they exist.** - **x-intercepts:** - A. The surface intersects the x-axis at \( x = 0 \). - B. There are no x-intercepts. - **y-intercepts:** - A. The surface intersects the y-axis at \( y = 0 \). - B. There are no y-intercepts. - **z-intercepts:** - A. The surface intersects the z-axis at \( z = 0 \). - B. There are no z-intercepts. **b. Find the equations of the xy-, xz-, and yz-traces, if they exist.** - **xy-trace:** - A. The trace is not a single point. The equation of the xy-trace is [blank]. - B. The trace is the single point \( (0, 0, 0) \). - C. There is no xy-trace. - **xz-trace:** - A. The trace is not a single point. The equation of the xz-trace is [blank]. - B. The trace is the single point \( (0, 0, 0) \). - C. There is no xz-trace. There is no information provided for the yz-trace. **c. Identify and sketch a graph of the surface.** The problem does not provide a sketch; however, based on the equation, the surface is indicative of an elliptic cone centered at the origin. This representation helps in understanding the geometrical shape and characteristics of the surface in a 3D coordinate system.
Expert Solution
Step 1: Solution 01

The given quadratic surface is x squared over 100 plus y squared over 25 equals z squared

(a)

Find the intercepts with three coordinate axes If they exist as follows

 x equals y equals 0, then 

table row cell 0 squared over 100 plus 0 squared over 25 end cell equals cell z squared end cell row cell z squared end cell equals 0 row z equals 0 end table

so, the intercept is origin i.e left parenthesis x comma y comma z right parenthesis equals left parenthesis 0 comma 0 comma 0 right parenthesis.

Put,x equals z equals 0 then 

Error converting from MathML to accessible text.

so, the intercept is origin i.e.left parenthesis x comma y comma z right parenthesis equals left parenthesis 0 comma 0 comma 0 right parenthesis.

Put,y equals z equals 0 then 

Error converting from MathML to accessible text.

so, the intercept is origin i.e.left parenthesis x comma y comma z right parenthesis equals left parenthesis 0 comma 0 comma 0 right parenthesis.


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