Exam 63: Rotation of Conics

Determine whether the statement is true or false? Justify your answer. ​ The graph of the equation $x ^ { 2 } + x y + k y ^ { 2 } + 12 x + 16 = 0$ where, $k$ is any constant less than $\frac { 1 } { 4 }$ ,is a hyperbola. ​

Select the graph of degenerate conic.​ $36 x ^ { 2 } - 72 x y + 36 y ^ { 2 } = 0$ ​

Select the graph of the following equation​ $36 x ^ { 2 } + 72 x y + 36 y ^ { 2 } = 0$ ​

Select the graph of degenerate conic.​ $y ^ { 2 } - 49 x ^ { 2 } = 0$ ​

Select the graph of the following equation.​ $x y + 4 = 0$ ​

Identify the conic by writing the equation in standard form.​ $25 x ^ { 2 } + 4 y ^ { 2 } - 400 x + 72 y + 1824 = 0$ ​

Use the Quadratic Formula to solve for $y$ .​ $x ^ { 2 } + 12 x y + 36 y ^ { 2 } - 5 x - y - 3 = 0$ ​

Use the discriminant to classify the graph.​ $100 x ^ { 2 } - 100 x y + 25 y ^ { 2 } + 12 y = 0$ ​

Rotate the axes to eliminate the xy-term in the equation.Then write the equation in standard form.​ $x y - 6 = 0$ ​

Determine the angle $\theta$ through which the axes are rotated.Round your answer to two decimal places.​ $x ^ { 2 } + 2 x y + y ^ { 2 } = 10$ ​

The $x ^ { \prime } y ^ { \prime }$ -coordinate system has been rotated $\theta$ degrees from the $x y$ -coordinate system.The coordinates of a point in the $x y$ -coordinate system are given.Find the coordinates of the point in the rotated coordinate system.​ $\theta = 90 ^ { \circ }$ , $( 2,2 )$ ​

Use the Quadratic Formula to solve for $y$ .​ $14 x ^ { 2 } - 10 x y + 9 y ^ { 2 } - 41 = 0$ ​

Use the discriminant to classify the graph.​ $36 x ^ { 2 } - 144 x y + y ^ { 2 } - 10 x + 5 y = 0$ ​

Select the graph of degenerate conic.​ $x ^ { 2 } - 14 x y + y ^ { 2 } = 0$ ​

Determine the angle $\theta$ through which the axes are rotated.Round your answer to two decimal places.​ $5 x ^ { 2 } - 10 x y + 12 y ^ { 2 } + ( 5 \sqrt { 11 } - 10 ) x - ( 7 \sqrt { 11 } + 11 ) y = 93$ ​

Use the Quadratic Formula to solve for $y$ .​ $x ^ { 2 } + x y + 4 y ^ { 2 } + x + y - 4 = 0$ ​

Determine the angle $\theta$ through which the axes are rotated.Round your answer to two decimal places.​ $17 x ^ { 2 } + 72 x y - 3 y ^ { 2 } = 77$ ​

The $x ^ { \prime } y ^ { \prime }$ -coordinate system has been rotated $\theta$ degrees from the $x y$ -coordinate system.The coordinates of a point in the $x y$ -coordinate system are given.Find the coordinates of the point in the rotated coordinate system.​ $\theta = 45 ^ { \circ }$ , $( 10,10 )$ ​

Use a graphing utility to graph the conic.Determine the angle $\theta$ through which the axes are rotated.​ $x ^ { 2 } + 2 x y + y ^ { 2 } = 25$ ​

Use the discriminant to classify the graph.​ $x ^ { 2 } + 6 x y + 9 y ^ { 2 } - 5 x - y - 3 = 0$ ​