How many of the following statements are correct ___

1. A conic with eccentricity equal to one is called a parabola

2. If $a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0$ represents a parabola, then

$a b c + 2 f g h - a f^{2} - b g^{2} - c h^{2} \neq 0 a n d h^{2} = a b$

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Solution

First statement is the definition of parabola. we call the conic with eccentricity equal to one, parabola.

The second statement is related to how to find if a given equation represents a parabola. To find that, we have to find $△ = a b c + 2 f g h - a f^{2} - b g^{2} - c h^{2}$

If $△ \neq 0 a n d h^{2} = a b$ , it represents a parabola.

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Similar questions

Q. 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}

is and

If the equation
$a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0$
represents a pair of straight lines, then
$a b c + 2 f g h - a f^{2} - b g^{2} + c h^{2} = 0$

Q. Assertion :The equation

3 x^{2} - 2 y^{2} + 4 x - 6 y = 0

represents a hyperbola. Reason: The second degree equation

a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0

represents a hyperbola if

a b c + 2 f g h - a f^{2} - b g^{2} - c h^{2} \neq 0

h^{2} > a b

Q. Assertion(A): The equation

(x + 1) (y - 3) = 4

represents a rectangular hyperbola.
Reason(R): The equation

a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0

represents a rectangular hyperbola if

a + b = 0, h^{2} > a b

and

a b c + 2 f g h - a f^{2} - b g^{2} - c h^{2} = 0

If the pair of lines $a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0$ intersect on the y axis, then

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How many of the following statements are correct ___ 1. A conic with eccentricity equal to one is called a parabola 2. If ax2+2hxy+by2+2gx+2fy+c=0 represents a parabola, then abc+2fgh−af2−bg2−ch2≠0 and h2=ab

How many of the following statements are correct ___

1. A conic with eccentricity equal to one is called a parabola

2. If $a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0$ represents a parabola, then

$a b c + 2 f g h - a f^{2} - b g^{2} - c h^{2} \neq 0 a n d h^{2} = a b$