Equation of the ellipse which passes through the point...

Equation of the ellipse which passes through the point $(-3,1)$, whose axes are the coordinate axes and has eccentricity $\sqrt{\frac{2}{5}}$ is
(A) $5 x^{2}+3 y^{2}-48=0$
(B) $3 x^{2}+5 y^{2}-15=0$
(C) $5 x^{2}+3 y^{2}-32=0$
(D) $3 x^{2}+5 y^{2}-32=0$

Key Concepts

Standard Equation of an Ellipse

This concept involves recognizing that an ellipse with its axes aligned with the coordinate axes has a standard equation of the form (x^2/a^2) + (y^2/b^2) = 1, where a and b are the lengths of the semi-major and semi-minor axes respectively. This form is crucial as it provides a baseline from which other properties of the ellipse, such as its eccentricity and its intersection with given points, can be derived.

Axes Aligned with the Coordinate Axes

When an ellipse’s axes are aligned with the coordinate axes, the equation simplifies by avoiding any cross-product terms (xy terms). This indicates that the ellipse is not rotated, which simplifies the identification of the semi-major and semi-minor axes and facilitates solving problems involving its geometry and algebraic properties.

Eccentricity of an Ellipse

Eccentricity (e) is a measure of how much an ellipse deviates from being circular. It is defined by the relationship e = ?(1 - (b^2/a^2)) for an ellipse with a > b. Eccentricity plays an essential role in characterizing the shape of the ellipse and is used to relate the geometry (the axes lengths) to other defining parameters of the ellipse.

Determining Conic Parameters Using a Point

This concept involves substituting the coordinates of a given point into the equation of the conic section in order to determine unknown parameters. By ensuring that the conic passes through a specific point, one can solve for constants or scaling factors, thereby obtaining the particular equation that satisfies both the geometric properties and the location constraint.

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