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From where does this equation: h^2-ab comes in conic sections and what does it represents?

From where does this equation: h^2-ab comes in conic sections and what does it represents?

Grade:12

1 Answers

Arun
25750 Points
5 years ago
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The equation ax2Β + 2hxy + by2Β + 2gx + 2fy + c = 0. Represents a second degree equation where a, h, b doesn’t variables simultaneously.
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Let a β‰  0.
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Now, the above equation becomes
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Β Β Β Β Β Β Β  a2Β x2Β + 2ax (hy + g) = aby2 – 2afy – ac
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on completing the square on the left side, we get,
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Β Β Β Β Β Β Β  a2Β x2Β + 2ax (hy + g) = y2Β (h2 – ab) + 2y (gh – af) + g2 – ac.
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i.e.Β Β Β  (ax + hy + g) =Β + √y2(h2–ab)+2y(gh–af) +(g2–ac)
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We cannot obtain x in terms of y, involving only terms of the first degree, unless the quantity under the radical sign be a perfect square. The condition for this is,
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(gh – af)2Β = (h2 – ab) (g2 – ac)
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hope it clears.
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Now if Delta is 0
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  1. h^2 – ab > 0 intersecting real lines
  2. h^2 – ab = 0. parallel lines
  3. h^2 – ab

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