Question

When the axis are rotated through an angle π/4, find the transformed equation of 3x² + 10xy + 3y² = 9.

Answer 1

Step-by-step explanation:

Given When the axis are rotated through an angle π/4, find the transformed equation of 3x² + 10xy + 3y² = 9.

• Now to form a new cartesian plane with coordinates (x1,y1), a conic equation of the form ax^2 + bxy + cy^2 + dx + ey + f = 0 is rotated by an angle theta.
• So the relation can be written as  
• So x = x1 cos theta – y1 sin theta  
• Or x1 = x cos theta + y sin theta
• Also y = x1 sin theta + y1 cos theta
• Or y1 = - x sin theta + y cos theta
• Now we have from the equation 3x^2 + 10 xy + 3y^2 – 9 = 0,
• a = 3, b = 10 and c = 3
• Now we need to get theta, so cot 2 theta = a – c / b
•                                                                         = 0 / 3  
•                                                                          = 0,
•                                                        So theta = π / 4
• Now we have the expression as x = x1 cos π/4 – y sin π/4
•                                          Also y = x1 sin π/4 + y1 cos π/4
• therefore x = x1/√2 – y1/√2 and y = x1/√2 + y1/√2
• so we get the equation as  
•  3(x1/√2 – y1/√2)^2 + 10 (x1/√2 – y1/√2)( x1/√2 + y1/√2) + 3(x1/√2 + y1/√2)^2 – 9 = 0
•      We can write it as  
•   3 (x1^2/2 + y1^2/2 – x1y1) + 10 (x1^2 / 2 – y1^2 / 2) + 3(x1^2 / 2 + y1^2 / 2 + x1y1) – 9 = 0
•  Simplifying we get
• 3x1^2 / 2 + 3x1^2 / 2 + 10 x1^2/2 – 3 x1y1 – 10y1^2/2 + 6y1^2/2 + 3x1y1 – 9 = 0
•                                     Or 8x1^2 – 2y1^2 – 9 = 0

Reference link will be

https://brainly.in/question/10688038

Answer 2

Answer:

Hlo Mate..

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