The eccentricity and the coordinates of a focus of the conic...

The eccentricity and the coordinates of a focus of the conic $(5 x-10)^{2}+(5 y+15)^{2}=(x-2 y+7)^{2}$ are given by
(a) $\sqrt{5},(2,-3)$
(b) $\frac{1}{\sqrt{5}},(2,-3)$
(c) $\frac{1}{\sqrt{5}},(3,-2)$
(d) $\frac{1}{\sqrt{5}},(3,2)$

Key Concepts

Eccentricity

Eccentricity is a parameter that measures how much a conic section deviates from being circular. It is defined as the ratio of the distance from any point on the conic to the focus over its distance to the directrix. The value of eccentricity determines the type of conic: for example, an eccentricity of 1 corresponds to a parabola, less than 1 to an ellipse, and greater than 1 to a hyperbola.

Distance Formula and Equation Normalization

The distance formula is widely used to express the distance between a point and a fixed point or between a point and a line. In the derivation of conic sections from the focus-directrix definition, proper normalization of the distance from a point to a line—by dividing by the square root of the sum of the squares of the line coefficients—is essential to correctly formulate the algebraic equation representing the conic.

Conic Sections

Conic sections are curves formed by the intersection of a plane with a double-napped cone. They include circles, ellipses, parabolas, and hyperbolas. Understanding the general quadratic representation of these curves helps in identifying and analyzing their geometric properties and behavior.

Focus-Directrix Property

The focus-directrix property defines a conic section as the set of points whose distances to a fixed point (the focus) and a fixed line (the directrix) are in a constant ratio. This property provides a geometric definition that is central to deriving the equation of a conic and finding its key features, such as the focus and the directrix.

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