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If two adjacent vertices of a parallelogram are (3,2) and (-1,0) and the diagonals intersect at (2, -5), then find the coordinates of the other two vertices. If α, β are zero of quadratic polynomial kx2 + 4x + 4, find the values of k such that (α + β)2 -2 αβ =24?
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If two adjacent vertices of a parallelogram are (3,2) and (-1,0) and t...
Coordinates of Other Two Vertices of Parallelogram
- Let the coordinates of the other two vertices be (x1, y1) and (x2, y2).
- Since opposite sides of a parallelogram are parallel, the vector joining the two adjacent vertices will be the same as the vector joining the other two vertices.
- So, (x1 - 3, y1 - 2) = (-1 - 3, 0 - 2) and (x2 - 3, y2 - 2) = (-1 - 3, 0 - 2).
- Solving these equations, we get x1 = -5, y1 = 0 and x2 = 1, y2 = -2.
- Therefore, the coordinates of the other two vertices are (-5, 0) and (1, -2).
Values of k for Quadratic Polynomial
- Given quadratic polynomial kx^2 + 4x + 4 = 0 has roots α and β.
- Sum of roots, α + β = -b/a = -4/k and product of roots, αβ = c/a = 4/k.
- Substituting these values in the given equation (α + β)^2 - 2αβ = 24, we get (-4/k)^2 - 2(4/k) = 24.
- Simplifying, we get 16/k^2 - 8/k - 24 = 0.
- Multiplying by k^2 to simplify further, we get 16 - 8k - 24k^2 = 0.
- Rearranging the terms, we get 24k^2 + 8k - 16 = 0.
- Dividing the equation by 8, we get 3k^2 + k - 2 = 0.
- Factoring the quadratic equation, we get (3k + 2)(k - 1) = 0.
- Therefore, the values of k that satisfy the given equation are k = -2/3 and k = 1.
- Let the coordinates of the other two vertices be (x1, y1) and (x2, y2).
- Since opposite sides of a parallelogram are parallel, the vector joining the two adjacent vertices will be the same as the vector joining the other two vertices.
- So, (x1 - 3, y1 - 2) = (-1 - 3, 0 - 2) and (x2 - 3, y2 - 2) = (-1 - 3, 0 - 2).
- Solving these equations, we get x1 = -5, y1 = 0 and x2 = 1, y2 = -2.
- Therefore, the coordinates of the other two vertices are (-5, 0) and (1, -2).
Values of k for Quadratic Polynomial
- Given quadratic polynomial kx^2 + 4x + 4 = 0 has roots α and β.
- Sum of roots, α + β = -b/a = -4/k and product of roots, αβ = c/a = 4/k.
- Substituting these values in the given equation (α + β)^2 - 2αβ = 24, we get (-4/k)^2 - 2(4/k) = 24.
- Simplifying, we get 16/k^2 - 8/k - 24 = 0.
- Multiplying by k^2 to simplify further, we get 16 - 8k - 24k^2 = 0.
- Rearranging the terms, we get 24k^2 + 8k - 16 = 0.
- Dividing the equation by 8, we get 3k^2 + k - 2 = 0.
- Factoring the quadratic equation, we get (3k + 2)(k - 1) = 0.
- Therefore, the values of k that satisfy the given equation are k = -2/3 and k = 1.
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If two adjacent vertices of a parallelogram are (3,2) and (-1,0) and the diagonals intersect at (2, -5), then find the coordinates of the other two vertices. If α, β are zero of quadratic polynomial kx2 + 4x + 4, find the values of k such that (α + β)2 -2 αβ =24?
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If two adjacent vertices of a parallelogram are (3,2) and (-1,0) and the diagonals intersect at (2, -5), then find the coordinates of the other two vertices. If α, β are zero of quadratic polynomial kx2 + 4x + 4, find the values of k such that (α + β)2 -2 αβ =24? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about If two adjacent vertices of a parallelogram are (3,2) and (-1,0) and the diagonals intersect at (2, -5), then find the coordinates of the other two vertices. If α, β are zero of quadratic polynomial kx2 + 4x + 4, find the values of k such that (α + β)2 -2 αβ =24? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If two adjacent vertices of a parallelogram are (3,2) and (-1,0) and the diagonals intersect at (2, -5), then find the coordinates of the other two vertices. If α, β are zero of quadratic polynomial kx2 + 4x + 4, find the values of k such that (α + β)2 -2 αβ =24?.
If two adjacent vertices of a parallelogram are (3,2) and (-1,0) and the diagonals intersect at (2, -5), then find the coordinates of the other two vertices. If α, β are zero of quadratic polynomial kx2 + 4x + 4, find the values of k such that (α + β)2 -2 αβ =24? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about If two adjacent vertices of a parallelogram are (3,2) and (-1,0) and the diagonals intersect at (2, -5), then find the coordinates of the other two vertices. If α, β are zero of quadratic polynomial kx2 + 4x + 4, find the values of k such that (α + β)2 -2 αβ =24? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If two adjacent vertices of a parallelogram are (3,2) and (-1,0) and the diagonals intersect at (2, -5), then find the coordinates of the other two vertices. If α, β are zero of quadratic polynomial kx2 + 4x + 4, find the values of k such that (α + β)2 -2 αβ =24?.
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