Rotate the Axes So That the New Equation Contains No $17 x ^ { 2 } - 12 x y + 8 y ^ { 2 } - 68 x + 24 y - 12 = 0$

question 15

Multiple Choice

Rotate the axes so that the new equation contains no xy-term. Discuss the new equation.
- $17 x ^ { 2 } - 12 x y + 8 y ^ { 2 } - 68 x + 24 y - 12 = 0$

• A) \(\theta = 26.6 ^ { \circ }\)
• \[\frac { x ^ { \prime 2 } } { 4 } + \frac { y ^ { \prime 2 } } { 16 } = 1\]
• ellipse
• center at \(( 0,0 )\)
• major axis is \(y ^ { \prime } -\) axis
• vertices at \(( 0 , \pm 4 )\)
• B) \(\theta = 63.4 ^ { \circ }\)
• \[\frac { \left( x ^ { \prime } - \frac { 2 \sqrt { 5 } } { 5 } \right) ^ { 2 } } { 16 } + \frac { \left( y ^ { \prime } + \frac { 4 \sqrt { 5 } } { 5 } \right) ^ { 2 } } { 4 } = 1\]
• ellipse
• center at \(\left( \frac { 2 \sqrt { 5 } } { 5 } , - \frac { 4 \sqrt { 5 } } { 5 } \right)\)
• major axis is \(x ^ { \prime }\)-axis
• vertices at \(\left( 4 + \frac { 2 \sqrt { 5 } } { 5 } , - \frac { 4 \sqrt { 5 } } { 5 } \right)\) and \(\left( - 4 + \frac { 2 \sqrt { 5 } } { 5 } , - \frac { 4 \sqrt { 5 } } { 5 } \right)\)
• C) \(\theta = 63.4 ^ { \circ }\)
• \(x ^ { \prime 2 } = - 16 y ^ { \prime }\)
• parabola
• vertex at \(( 0,0 )\)
• focus at \(( 0 , - 4 )\)
• D) \(\theta = 63.4 ^ { \circ }\)
• \[\frac { x ^ { \prime 2 } } { 16 } - \frac { y ^ { \prime 2 } } { 4 } = 1\]
• hyperbola
• center at \(( 0,0 )\)
• transverse axis is the \(x ^ { \prime }\)-axis
• vertices at \(( \pm 4,0 )\)

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Definitions:

Variable Overhead Rate Variance

The difference between the actual variable overhead incurred and the expected variable overhead based on standard cost.

Variable Manufacturing Overhead

Costs in the manufacturing process that change with the level of production output, such as utilities and materials used in production.

Last Month

Refers to the period of time from the first to the last day of the month immediately preceding the current month.

Variable Overhead Efficiency Variance

The difference between the actual hours taken to produce goods and the standard hours expected, multiplied by the variable overhead rate.