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The vertices of a triangle are A(x1,x1tana), B(x2, x2tanb) and C(x3,x3tanc). if tye circumcentre of triangle ABC coincides with the origin and H(m,n) is the orthocentre, find n/m?
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The vertices of a triangle are A(x1,x1tana), B(x2, x2tanb) and C(x3,x3...
Given Information:
- Vertices of the triangle: A(x1, x1tanα), B(x2, x2tanβ), C(x3, x3tanγ)
- Circumcentre coincides with the origin
- Orthocentre H(m, n)
Approach:
- Circumcentre of a triangle is the point of concurrency of the perpendicular bisectors of the sides.
- Orthocentre of a triangle is the point of concurrency of the altitudes.
- As the circumcentre coincides with the origin, it is the centroid of the triangle.
- The centroid divides each altitude in a ratio of 2:1, with the longer segment being closer to the vertex.
Finding the Centroid (Origin):
- Using the coordinates of the vertices A, B, and C, find the coordinates of the centroid (0, 0).
Finding the Orthocentre H(m, n):
- The orthocentre is the point where the altitudes of the triangle intersect.
- The altitude from vertex A passes through the midpoint of side BC.
- Using the coordinates of A, B, and C, find the equation of the altitude from A.
- Find the intersection point of these altitudes to determine the orthocentre H(m, n).
Calculating n/m:
- Once you have the coordinates of the orthocentre H(m, n), calculate the ratio n/m.
Conclusion:
- The ratio n/m represents the coordinates of the orthocentre H(m, n) in the triangle ABC with circumcentre at the origin.
- Vertices of the triangle: A(x1, x1tanα), B(x2, x2tanβ), C(x3, x3tanγ)
- Circumcentre coincides with the origin
- Orthocentre H(m, n)
Approach:
- Circumcentre of a triangle is the point of concurrency of the perpendicular bisectors of the sides.
- Orthocentre of a triangle is the point of concurrency of the altitudes.
- As the circumcentre coincides with the origin, it is the centroid of the triangle.
- The centroid divides each altitude in a ratio of 2:1, with the longer segment being closer to the vertex.
Finding the Centroid (Origin):
- Using the coordinates of the vertices A, B, and C, find the coordinates of the centroid (0, 0).
Finding the Orthocentre H(m, n):
- The orthocentre is the point where the altitudes of the triangle intersect.
- The altitude from vertex A passes through the midpoint of side BC.
- Using the coordinates of A, B, and C, find the equation of the altitude from A.
- Find the intersection point of these altitudes to determine the orthocentre H(m, n).
Calculating n/m:
- Once you have the coordinates of the orthocentre H(m, n), calculate the ratio n/m.
Conclusion:
- The ratio n/m represents the coordinates of the orthocentre H(m, n) in the triangle ABC with circumcentre at the origin.
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The vertices of a triangle are A(x1,x1tana), B(x2, x2tanb) and C(x3,x3tanc). if tye circumcentre of triangle ABC coincides with the origin and H(m,n) is the orthocentre, find n/m?
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The vertices of a triangle are A(x1,x1tana), B(x2, x2tanb) and C(x3,x3tanc). if tye circumcentre of triangle ABC coincides with the origin and H(m,n) is the orthocentre, find n/m? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The vertices of a triangle are A(x1,x1tana), B(x2, x2tanb) and C(x3,x3tanc). if tye circumcentre of triangle ABC coincides with the origin and H(m,n) is the orthocentre, find n/m? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The vertices of a triangle are A(x1,x1tana), B(x2, x2tanb) and C(x3,x3tanc). if tye circumcentre of triangle ABC coincides with the origin and H(m,n) is the orthocentre, find n/m?.
The vertices of a triangle are A(x1,x1tana), B(x2, x2tanb) and C(x3,x3tanc). if tye circumcentre of triangle ABC coincides with the origin and H(m,n) is the orthocentre, find n/m? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The vertices of a triangle are A(x1,x1tana), B(x2, x2tanb) and C(x3,x3tanc). if tye circumcentre of triangle ABC coincides with the origin and H(m,n) is the orthocentre, find n/m? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The vertices of a triangle are A(x1,x1tana), B(x2, x2tanb) and C(x3,x3tanc). if tye circumcentre of triangle ABC coincides with the origin and H(m,n) is the orthocentre, find n/m?.
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