How many of the following statements are correct: A conic with eccentricity equal to one is called a parabola. If a x^{2}+2 h x y+b y^{2}+2 g x+2 f y+c=0 represents a parabola, then a b c+2 f g h-a f^{2}-b g^{2}-c h^{2} does not equal 0 and h^{2}=a b.
Question
How many of the following statements are correct:
- A conic with eccentricity equal to one is called a parabola.
- If $a x^{2}+2 h x y+b y^{2}+2 g x+2 f y+c=0$ represents a parabola, then $a b c+2 f g h-a f^{2}-b g^{2}-c h^{2} \neq 0$ and $h^{2}=a b$.
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Answer
1. First Statement: The first statement correctly defines a parabola. A conic section with an eccentricity equal to one is known as a parabola.
2. Second Statement: The second statement provides a method to verify if a given equation represents a parabola. The given equation is:
$$ a x^{2}+2 h x y+b y^{2}+2 g x+2 f y+c=0 $$
To determine if this equation represents a parabola, we need to compute:
$$ \Delta = a b c + 2 f g h - a f^{2} - b g^{2} - c h^{2} $$
For the equation to be identified as a parabola, the following conditions must be met:
- (\Delta \neq 0)
- (h^{2} = a b)
Both conditions must be satisfied for the equation to represent a parabola.
Thus, both statements are correct.
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