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The base of triangle is passing through a fixed point (a,b) and its sides are bisected at right angles by lines, y2 - 4xy - 5x2 = 0, then the locus of the vertex of triangle is
- a)2x2 + 2y2 + (3a + 2b)x + (2a - 3b)y = 0
- b)2x2 + 2y2 + (3a + 2b)x + (2a + 3b)y = 0
- c)2x2 - 2h2 + (3a + 2b)x + (2a + 3b)y = 0
- d)None of these
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
The base of triangle is passing through a fixed point (a,b) and its si...
Locus of the Vertex of Triangle
Given:
- The base of triangle is passing through a fixed point (a,b).
- Its sides are bisected at right angles by lines, y2 - 4xy - 5x2 = 0.
To find:
- The locus of the vertex of the triangle.
Approach:
- Let the coordinates of the vertex be (h,k).
- Use the perpendicular bisectors of the sides to find the coordinates of the circumcenter of the triangle.
- Use the fact that the circumcenter lies on the perpendicular bisectors of the sides to find an equation for the locus of the circumcenter.
- Use the fact that the circumcenter and the vertex are equidistant from the endpoints of the base to find an equation for the locus of the vertex.
- Equate the two equations to find the locus of the vertex.
Solution:
Finding the Circumcenter:
- The equation of the perpendicular bisector of the line segment joining (a,b) and (0,0) is x - (a/2) + y - (b/2) = 0.
- The equation of the perpendicular bisector of the line segment joining (a,b) and (5/4,1) is y + 2x - 5/2 = 0.
- The point of intersection of these two lines is the circumcenter of the triangle.
- Solving the equations, we get the coordinates of the circumcenter as (h,k) = (5a/8 + b/4, a/4 + b/2).
Finding the Locus of the Circumcenter:
- The circumcenter of a triangle is equidistant from the three vertices of the triangle.
- Therefore, the circumcenter lies on the perpendicular bisectors of the sides of the triangle.
- The equation of the perpendicular bisector of the line segment joining (a,b) and (0,0) is x - (a/2) + y - (b/2) = 0.
- The equation of the perpendicular bisector of the line segment joining (a,b) and (5/4,1) is y + 2x - 5/2 = 0.
- Using the coordinates of the circumcenter, we can find an equation for the locus of the circumcenter.
- Substituting the coordinates of the circumcenter into the equations of the perpendicular bisectors, we get two equations:
- 5x - 6y + 5a = 0
- 6x + 5y - 8a - 4b = 0
- Simplifying these equations, we get:
- 25x^2 + 36y^2 - 60ax + 72by - 25a^2 = 0
- 36x^2 + 25y^2 - 96ax - 80by + 64a^2 + 64ab = 0
- Multiplying the first equation by 16 and the second equation by 9, we get:
- 400x^2 + 576y^2 - 960ax + 1152by - 400a^2 = 0
- 324x^2 + 225y^2 - 864ax - 720by + 576a^2 + 576ab = 0
-
Given:
- The base of triangle is passing through a fixed point (a,b).
- Its sides are bisected at right angles by lines, y2 - 4xy - 5x2 = 0.
To find:
- The locus of the vertex of the triangle.
Approach:
- Let the coordinates of the vertex be (h,k).
- Use the perpendicular bisectors of the sides to find the coordinates of the circumcenter of the triangle.
- Use the fact that the circumcenter lies on the perpendicular bisectors of the sides to find an equation for the locus of the circumcenter.
- Use the fact that the circumcenter and the vertex are equidistant from the endpoints of the base to find an equation for the locus of the vertex.
- Equate the two equations to find the locus of the vertex.
Solution:
Finding the Circumcenter:
- The equation of the perpendicular bisector of the line segment joining (a,b) and (0,0) is x - (a/2) + y - (b/2) = 0.
- The equation of the perpendicular bisector of the line segment joining (a,b) and (5/4,1) is y + 2x - 5/2 = 0.
- The point of intersection of these two lines is the circumcenter of the triangle.
- Solving the equations, we get the coordinates of the circumcenter as (h,k) = (5a/8 + b/4, a/4 + b/2).
Finding the Locus of the Circumcenter:
- The circumcenter of a triangle is equidistant from the three vertices of the triangle.
- Therefore, the circumcenter lies on the perpendicular bisectors of the sides of the triangle.
- The equation of the perpendicular bisector of the line segment joining (a,b) and (0,0) is x - (a/2) + y - (b/2) = 0.
- The equation of the perpendicular bisector of the line segment joining (a,b) and (5/4,1) is y + 2x - 5/2 = 0.
- Using the coordinates of the circumcenter, we can find an equation for the locus of the circumcenter.
- Substituting the coordinates of the circumcenter into the equations of the perpendicular bisectors, we get two equations:
- 5x - 6y + 5a = 0
- 6x + 5y - 8a - 4b = 0
- Simplifying these equations, we get:
- 25x^2 + 36y^2 - 60ax + 72by - 25a^2 = 0
- 36x^2 + 25y^2 - 96ax - 80by + 64a^2 + 64ab = 0
- Multiplying the first equation by 16 and the second equation by 9, we get:
- 400x^2 + 576y^2 - 960ax + 1152by - 400a^2 = 0
- 324x^2 + 225y^2 - 864ax - 720by + 576a^2 + 576ab = 0
-
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The base of triangle is passing through a fixed point (a,b) and its si...
Option A is correct.
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The base of triangle is passing through a fixed point (a,b) and its sides are bisected at right angles by lines, y2 - 4xy - 5x2 = 0, then the locus of the vertex of triangle isa)2x2 + 2y2 + (3a + 2b)x + (2a - 3b)y = 0b)2x2 + 2y2 + (3a + 2b)x + (2a + 3b)y = 0c)2x2 - 2h2 + (3a + 2b)x + (2a + 3b)y = 0d)None of theseCorrect answer is option 'A'. Can you explain this answer?
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The base of triangle is passing through a fixed point (a,b) and its sides are bisected at right angles by lines, y2 - 4xy - 5x2 = 0, then the locus of the vertex of triangle isa)2x2 + 2y2 + (3a + 2b)x + (2a - 3b)y = 0b)2x2 + 2y2 + (3a + 2b)x + (2a + 3b)y = 0c)2x2 - 2h2 + (3a + 2b)x + (2a + 3b)y = 0d)None of theseCorrect answer is option 'A'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The base of triangle is passing through a fixed point (a,b) and its sides are bisected at right angles by lines, y2 - 4xy - 5x2 = 0, then the locus of the vertex of triangle isa)2x2 + 2y2 + (3a + 2b)x + (2a - 3b)y = 0b)2x2 + 2y2 + (3a + 2b)x + (2a + 3b)y = 0c)2x2 - 2h2 + (3a + 2b)x + (2a + 3b)y = 0d)None of theseCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The base of triangle is passing through a fixed point (a,b) and its sides are bisected at right angles by lines, y2 - 4xy - 5x2 = 0, then the locus of the vertex of triangle isa)2x2 + 2y2 + (3a + 2b)x + (2a - 3b)y = 0b)2x2 + 2y2 + (3a + 2b)x + (2a + 3b)y = 0c)2x2 - 2h2 + (3a + 2b)x + (2a + 3b)y = 0d)None of theseCorrect answer is option 'A'. Can you explain this answer?.
The base of triangle is passing through a fixed point (a,b) and its sides are bisected at right angles by lines, y2 - 4xy - 5x2 = 0, then the locus of the vertex of triangle isa)2x2 + 2y2 + (3a + 2b)x + (2a - 3b)y = 0b)2x2 + 2y2 + (3a + 2b)x + (2a + 3b)y = 0c)2x2 - 2h2 + (3a + 2b)x + (2a + 3b)y = 0d)None of theseCorrect answer is option 'A'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The base of triangle is passing through a fixed point (a,b) and its sides are bisected at right angles by lines, y2 - 4xy - 5x2 = 0, then the locus of the vertex of triangle isa)2x2 + 2y2 + (3a + 2b)x + (2a - 3b)y = 0b)2x2 + 2y2 + (3a + 2b)x + (2a + 3b)y = 0c)2x2 - 2h2 + (3a + 2b)x + (2a + 3b)y = 0d)None of theseCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The base of triangle is passing through a fixed point (a,b) and its sides are bisected at right angles by lines, y2 - 4xy - 5x2 = 0, then the locus of the vertex of triangle isa)2x2 + 2y2 + (3a + 2b)x + (2a - 3b)y = 0b)2x2 + 2y2 + (3a + 2b)x + (2a + 3b)y = 0c)2x2 - 2h2 + (3a + 2b)x + (2a + 3b)y = 0d)None of theseCorrect answer is option 'A'. Can you explain this answer?.
Solutions for The base of triangle is passing through a fixed point (a,b) and its sides are bisected at right angles by lines, y2 - 4xy - 5x2 = 0, then the locus of the vertex of triangle isa)2x2 + 2y2 + (3a + 2b)x + (2a - 3b)y = 0b)2x2 + 2y2 + (3a + 2b)x + (2a + 3b)y = 0c)2x2 - 2h2 + (3a + 2b)x + (2a + 3b)y = 0d)None of theseCorrect answer is option 'A'. Can you explain this answer? in English & in Hindi are available as part of our courses for JEE.
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Here you can find the meaning of The base of triangle is passing through a fixed point (a,b) and its sides are bisected at right angles by lines, y2 - 4xy - 5x2 = 0, then the locus of the vertex of triangle isa)2x2 + 2y2 + (3a + 2b)x + (2a - 3b)y = 0b)2x2 + 2y2 + (3a + 2b)x + (2a + 3b)y = 0c)2x2 - 2h2 + (3a + 2b)x + (2a + 3b)y = 0d)None of theseCorrect answer is option 'A'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
The base of triangle is passing through a fixed point (a,b) and its sides are bisected at right angles by lines, y2 - 4xy - 5x2 = 0, then the locus of the vertex of triangle isa)2x2 + 2y2 + (3a + 2b)x + (2a - 3b)y = 0b)2x2 + 2y2 + (3a + 2b)x + (2a + 3b)y = 0c)2x2 - 2h2 + (3a + 2b)x + (2a + 3b)y = 0d)None of theseCorrect answer is option 'A'. Can you explain this answer?, a detailed solution for The base of triangle is passing through a fixed point (a,b) and its sides are bisected at right angles by lines, y2 - 4xy - 5x2 = 0, then the locus of the vertex of triangle isa)2x2 + 2y2 + (3a + 2b)x + (2a - 3b)y = 0b)2x2 + 2y2 + (3a + 2b)x + (2a + 3b)y = 0c)2x2 - 2h2 + (3a + 2b)x + (2a + 3b)y = 0d)None of theseCorrect answer is option 'A'. Can you explain this answer? has been provided alongside types of The base of triangle is passing through a fixed point (a,b) and its sides are bisected at right angles by lines, y2 - 4xy - 5x2 = 0, then the locus of the vertex of triangle isa)2x2 + 2y2 + (3a + 2b)x + (2a - 3b)y = 0b)2x2 + 2y2 + (3a + 2b)x + (2a + 3b)y = 0c)2x2 - 2h2 + (3a + 2b)x + (2a + 3b)y = 0d)None of theseCorrect answer is option 'A'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice The base of triangle is passing through a fixed point (a,b) and its sides are bisected at right angles by lines, y2 - 4xy - 5x2 = 0, then the locus of the vertex of triangle isa)2x2 + 2y2 + (3a + 2b)x + (2a - 3b)y = 0b)2x2 + 2y2 + (3a + 2b)x + (2a + 3b)y = 0c)2x2 - 2h2 + (3a + 2b)x + (2a + 3b)y = 0d)None of theseCorrect answer is option 'A'. Can you explain this answer? tests, examples and also practice JEE tests.
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