The sum of the squares of the eccentricities of the conics (x^2)/(4)+(y^2)/(3)=1 and (x^2)/(4)-(

step 1 In this problem we have to find the sum of squares of the eccentricities of these two conic sect

The sum of the squares of the eccentricities of the conics...

The sum of the squares of the eccentricities of the conics $\frac{x^{2}}{4}+\frac{y^{2}}{3}=1$ and $\frac{x^{2}}{4}-\frac{y^{2}}{3}=1$ is
(a) 2
(b) $\sqrt{2}$
(c) $\frac{\sqrt{7}}{\sqrt{3}}$
(d) $\sqrt{7}$

Key Concepts

Hyperbola

A hyperbola is a conic section formed by the difference of distances from any point on the hyperbola to its two foci being constant. Its standard form helps identify key parameters such as the transverse and conjugate axes, and the eccentricity, which indicates the shape's divergence from a circle, is computed using these parameters.

Eccentricity

Eccentricity is a numerical measure that describes how much a conic section deviates from being circular. For an ellipse, it is defined based on the relationship between the semi-major and semi-minor axes, while for a hyperbola, it is calculated using the distances between the center and the vertices along with the conjugate axis. The concept of eccentricity is crucial for distinguishing between different conic sections and understanding their geometric properties.

Ellipse

An ellipse is a type of conic section defined by the sum of the distances from any point on the ellipse to its two foci being constant. Its standard equation helps determine its semi-major and semi-minor axes, and the eccentricity (a measure of how much the ellipse deviates from being circular) is calculated using these axes.

Conic Sections

Conic sections are curves obtained by intersecting a right circular cone with a plane. The main types of conic sections include circles, ellipses, parabolas, and hyperbolas. Each type has a standard equation form and unique geometric properties, which are central to the study of analytic geometry.

Standard Forms of Conics

The equations of conic sections are often expressed in their standard forms to clearly identify the type of conic and its key parameters. For instance, the ellipse and hyperbola have standard forms that reveal the lengths of their axes and centers, helping in determining properties like eccentricity and focal points.

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