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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
25763 Points
2 years ago
 
 
The equation ax2 + 2hxy + by2 + 2gx + 2fy + c = 0. Represents a second degree equation where a, h, b doesn’t variables simultaneously.
 
Let a ≠ 0.
 
Now, the above equation becomes
 
        a2 x2 + 2ax (hy + g) = aby2 – 2afy – ac
 
on completing the square on the left side, we get,
 
        a2 x2 + 2ax (hy + g) = y2 (h2 – ab) + 2y (gh – af) + g2 – ac.
 
i.e.    (ax + hy + g) = + √y2(h2–ab)+2y(gh–af) +(g2–ac)
 
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,
 
(gh – af)2 = (h2 – ab) (g2 – ac)
 
hope it clears.
 
Now if Delta is 0
 
  1. h^2 – ab > 0 intersecting real lines
  2. h^2 – ab = 0. parallel lines
  3. h^2 – ab

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