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The line passing through the extremely A of the major axis and extremity B of the minor axis of the ellipse x2 + 9y2 = 9 meets its auxiliary circle at the point M . Then the area of the triangle with vertices at A,M and the origin O is
- a)31/10
- b)29/10
- c)21/10
- d)27/10
Correct answer is option 'D'. Can you explain this answer?
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The line passing through the extremely A of the major axis and extremi...
Equation of line AM is x + 3y - 3 =0
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The line passing through the extremely A of the major axis and extremi...
Explanation:
The given ellipse equation is x^2 + 9y^2 = 9, which can be written as x^2/9 + y^2/1 = 1. This ellipse has a major axis of length 6 (2a = 6) and a minor axis of length 2 (2b = 2).
Finding the Endpoints of Major and Minor Axes:
- The endpoints of the major axis are (3, 0) and (-3, 0) denoted by points A and A'.
- The endpoints of the minor axis are (0, 1) and (0, -1) denoted by points B and B'.
Finding the Point of Intersection with Auxiliary Circle:
- The auxiliary circle of the ellipse has the equation x^2 + y^2 = 1.
- Substituting the equation of the ellipse into the auxiliary circle equation, we get x^2 + 9y^2 = 9 = x^2 + y^2.
- Solving the above equation gives y^2 = 1/10, which implies y = ±1/√10.
- Therefore, the points of intersection are (3, 1/√10) and (3, -1/√10) denoted by M and M'.
Calculating the Area of the Triangle:
- The area of the triangle with vertices A, M, and the origin O can be calculated using the formula for the area of a triangle given by half the magnitude of the cross product of vectors AM and AO.
- The vector AM is (3, 1/√10) and the vector AO is (-3, 0).
- Calculating the cross product magnitude gives |AM x AO| = |3(0) - (1/√10)(-3)| = 3/√10.
- Therefore, the area of the triangle is 1/2 * base * height = 1/2 * 3 * 3/√10 = 9/2√10 = 9√10/20 = 9/10.
- So, the correct answer is option 'D' (27/10).
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The line passing through the extremely A of the major axis and extremi...
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The line passing through the extremely A of the major axis and extremity B of the minor axis of the ellipse x2 + 9y2= 9 meets its auxiliary circle at the point M . Then the area of the triangle with vertices at A,M and the origin O isa)31/10b)29/10c)21/10d)27/10Correct answer is option 'D'. Can you explain this answer?
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The line passing through the extremely A of the major axis and extremity B of the minor axis of the ellipse x2 + 9y2= 9 meets its auxiliary circle at the point M . Then the area of the triangle with vertices at A,M and the origin O isa)31/10b)29/10c)21/10d)27/10Correct answer is option 'D'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The line passing through the extremely A of the major axis and extremity B of the minor axis of the ellipse x2 + 9y2= 9 meets its auxiliary circle at the point M . Then the area of the triangle with vertices at A,M and the origin O isa)31/10b)29/10c)21/10d)27/10Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The line passing through the extremely A of the major axis and extremity B of the minor axis of the ellipse x2 + 9y2= 9 meets its auxiliary circle at the point M . Then the area of the triangle with vertices at A,M and the origin O isa)31/10b)29/10c)21/10d)27/10Correct answer is option 'D'. Can you explain this answer?.
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