MATH SOLVE

2 months ago

Q:
# Identify the graph of the equation. What is the angle of rotation for the equation? (Picture provided)

Accepted Solution

A:

Answer:The equation is that of ellipse withe angle of rotation 30° ⇒ answer (d)Step-by-step explanation:* Lets talk about the general form of the conic equations- Ax² + Bxy + Cy² +D = 0 (center is the origin)- A is the coefficient of x² , B is the coefficient of xy C is the coefficient of y² , D is the numerical term* Now we will study how to know the type of the graph of this equation- If A and C have different signs (different values)∴ The equation is that of an ellipse- If A and C have different signs (different values)∴ The equation is on a hyperbola* Now look at the equation: 13x² + 6√3 xy + 7y² - 16 = 0∵ A = 13 , B = 6√3 , C = 7 , D = -16∵ A and C have same sign∴ The equation is that of an ellipse* Now lets find the angle of rotation by using the Rule:- tan(2Ф) = B/(A - C) ⇒ Ф is the angle of rotation- By using the value of A , B and C∴ tan(2Ф) = 6√3/(13 - 7) = 6√3/6 = √3∴ 2Ф = [tex]tan^{-1}\sqrt{3}=60[/tex]∴ 2Ф = 60° ⇒ divide both sides by 2∴ Ф = 30°∴ The angle of rotation is 30°∴ The equation is that of ellipse withe angle of rotation 30°* The graph represent the ellipse- The purple line represents the angle of rotation