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:

Simple Fruits

Fruits that develop from a single ovary of one flower and can be either fleshy, like cherries and tomatoes, or dry, like nuts and grains.

Seed Coat

The protective outer layer of a seed, which encases the embryonic plant and its nutrient store.

Endosperm

The nutrient-rich tissue inside seeds that provides nourishment to the developing plant embryo.

Embryo

An early stage of development in multicellular organisms after fertilization but before birth or hatching.