Each question contains Statement- 1 and Statement- 2 and has...

Each question contains Statement- 1 and Statement- 2 and has the following choices (a), (b), (c) and (d), out of which ONLY ONE is correct. (a) Statement- 1 is True, Statement- 2 is True; Statement- 2 is a correct explanation for Statement-1 (b) Statement- 1 is True, Statement- 2 is True; Statement- 2 is NOT a correct explanation for Statement-1 (c) Statement- 1 is True, Statement- 2 is False (d) Statement- 1 is False, Statement- 2 is True Statement 1 It is not possible to draw on a plane, a hyperbola and an ellipse having same foci. and Statement 2 $a x^{2}+2 h x y+b y^{2}+2 g x+2 f y+c=0$ represents a conic with eccentricity e, then for a hyperbola $e>1$ and for an ellipse $\mathrm{e}<1 .$

Each question contains Statement- 1 and Statement- 2 and has the following choices (a), (b), (c) and (d), out of which ONLY ONE is correct.
(a) Statement- 1 is True, Statement- 2 is True; Statement- 2 is a correct explanation for Statement-1
(b) Statement- 1 is True, Statement- 2 is True; Statement- 2 is NOT a correct explanation for Statement-1
(c) Statement- 1 is True, Statement- 2 is False
(d) Statement- 1 is False, Statement- 2 is True
Statement 1
It is not possible to draw on a plane, a hyperbola and an ellipse having same foci.
and
Statement 2 $a x^{2}+2 h x y+b y^{2}+2 g x+2 f y+c=0$ represents a conic with eccentricity e, then for a hyperbola $e>1$ and for an ellipse $\mathrm{e}<1 .$

Key Concepts

General Conic Equation

The general quadratic equation ax² + 2hxy + by² + 2gx + 2fy + c = 0 represents all conic sections depending on the values of the coefficients. The classification into ellipse, hyperbola, or parabola is determined by the discriminant (h² - ab) among other conditions. This equation is a unifying framework for studying the diverse properties and relations between different conics.

Eccentricity

Eccentricity is a parameter that describes the degree of deviation of a conic section from being circular. It is denoted by 'e' and helps distinguish between different types of conics: for ellipses, 0 ? e < 1 (with 0 representing a circle), for parabolas e = 1, and for hyperbolas e > 1. This concept is fundamental in understanding the geometric behavior and formulas of conic sections.

Conic Sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone. They include circles, ellipses, parabolas, and hyperbolas, each defined by specific geometric properties and equations. Their classification depends on the angle of the intersection, and they have significant applications in physics, astronomy, and engineering.

Foci

Foci are specific fixed points used in the definition of conic sections. For ellipses and hyperbolas, the sum or difference of the distances from any point on the curve to the two foci remains constant. The positioning of the foci is crucial in determining the shape and properties of the conic section, such as its eccentricity.

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