Rotate the Axes to Eliminate The $x y$ -Term in the Equation

question 95

Multiple Choice

Rotate the axes to eliminate the $x y$ -term in the equation. Then write the equation in standard form.
$$113 x ^ { 2 } - 30 x y + 113 y ^ { 2 } - 6272 = 0$$

• A) \(\frac { \left( x ^ { \prime } \right) ^ { 2 } } { 81 } + \frac { \left( y ^ { \prime } \right) ^ { 2 } } { 64 } = 1\)
• B) \(\frac { \left( x ^ { \prime } \right) ^ { 2 } } { 49 } + \frac { \left( y ^ { \prime } \right) ^ { 2 } } { 36 } = 1\)
• C) \(\frac { \left( x ^ { \prime } \right) ^ { 2 } } { 64 } + \frac { \left( y ^ { \prime } \right) ^ { 2 } } { 64 } = 1\)
• D) \(\frac { \left( x ^ { \prime } \right) ^ { 2 } } { 49 } + \frac { \left( y ^ { \prime } \right) ^ { 2 } } { 49 } = 1\)
• E) \(\frac { \left( x ^ { \prime } \right) ^ { 2 } } { 64 } + \frac { \left( y ^ { \prime } \right) ^ { 2 } } { 49 } = 1\)

---

Definitions:

Urban Areas

Regions characterized by high human population density and vast human-built features in comparison to areas surrounding them, typically cities and towns.

Immigration

The action of moving to a foreign country or region to live, often permanently.

Urban Growth

The expansion of cities and towns as a result of population increase and infrastructural development, often impacting the surrounding environment and agricultural lands.

Developing Countries

Countries characterized by less industrialization, a lower quality of life, and typically lower scores on the Human Development Index (HDI) than those seen in developed nations.