Rotate the Axes So That the New Equation Contains No $$x ^ { 2 } + x y + y ^ { 2 } - 3 y - 6 = 0$$

question 100

Multiple Choice

Rotate the axes so that the new equation contains no xy-term. Discuss the new equation.
- $$x ^ { 2 } + x y + y ^ { 2 } - 3 y - 6 = 0$$

• A)
• \(\begin{array}{l}
• \theta=45^{\circ} \\
• y^{\prime 2}=-18 x^{\prime} \\
• \text { parabola } \\
• \text { vertex at }(0,0) \\
• \text { focus at }\left(-\frac{9}{2}, 0\right)
• \end{array}\)
• B)
• \[
• \frac{\theta=45^{\circ}}{\left(x^{\prime}-\frac{\sqrt{2}}{2}\right)^{2}}+\frac{\left(y^{\prime}-\frac{3 \sqrt{2}}{2}\right)^{2}}{15}=1
• \]
• ellipse center at \( \left(\frac{\sqrt{2}}{2}, \frac{3 \sqrt{2}}{2}\right) \)
• major axis is \( y^{\prime} \)-axis vertices at \( \left(\frac{\sqrt{2}}{2},-\frac{3 \sqrt{2}}{2}\right) \) and \( \left(\frac{\sqrt{2}}{2}, \frac{9 \sqrt{2}}{2}\right) \)
• C) \[
• \theta=45^{\circ}
• \]
• \[
• \frac{x^{\prime 2}}{6}-\frac{y^{\prime 2}}{8}=1
• \]
• hyperbola
• center at \( (0,0) \)
• transverse axis is the \( \mathrm{x}^{\prime} \)-axis
• vertices at \( (\pm \sqrt{6}, 0) \)
• D)\[
• \begin{array}{l}
• \theta=45^{\circ} \\
• \frac{x^{\prime 2}}{3}+\frac{y^{2}}{4}=1
• \end{array}
• \]
• ellipse
• center at \( (0,0) \)
• major axis is \( y^{\prime} \)-axis
• vertices at \( (0, \pm 2) \)

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Definitions:

Evolutionary Advantage

A trait that increases an organism's likelihood of survival and reproduction in a particular environment.

Haploid

Referring to a cell or nucleus containing a single set of unpaired chromosomes, as seen in gametes (sperm and egg cells) of sexually reproducing organisms.

Meiosis

is a type of cell division that reduces the chromosome number by half, creating four haploid cells, each genetically distinct from the parent cell.

Chromosome Segregation

The process during cell division in which chromosomes are equally distributed into two new nuclei, ensuring each daughter cell receives an identical set of chromosomes.