Question

The eccentricity of an ellipse, with its centre at the origin, is \((1 / 2)\). If one of the directories is \(\mathrm{x}=4\), then equation of an ellipse is (a) \(3 \mathrm{x}^{2}+4 \mathrm{y}^{2}=1\) (b) \(3 x^{2}+4 y^{2}=12\) (c) \(4 x^{2}+3 y^{2}=12\) (d) \(4 x^{2}+3 y^{2}=1\)

Short answer

The short answer: Using the given eccentricity \(\frac{1}{2}\) and directrix \(x = 4\), we find the equation of the ellipse to be \(4x^2 + 3y^2 = 12\). Answer choice (c) is the correct option.

Step-by-step solution

Relationship between semi-major axis and directrix

Since the directrix is given by \(x=4\), the axis is parallel to the y-axis. Hence, the ellipse is of the form \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\), where \(a\) and \(b\) are the semi-major and semi-minor axes, respectively. For an ellipse, the distance from a focus point to the directrix is \(\frac{a}{e}\), where \(a\) is the semi-major axis and \(e\) is the eccentricity. Given that the eccentricity is \(\frac{1}{2}\) and that the distance between the origin to the directrix is \(4\), we can set up the following equation: \(\frac{a}{\frac{1}{2}} = 4\) Solving for \(a\), we get: \(a = 4 \cdot \frac{1}{2}\) \(a = 2\) So, the value of the semi-major axis \(a\) is \(2\).

Relationship between semi-major axis and semi-minor axis

Since eccentricity is given by \(e=\sqrt{1-\frac{b^2}{a^2}}\), and we now know the value of \(a\) and \(e\), we can find the value of \(b\). Plug in the given values of \(a\) and \(e\): \(\frac{1}{2}=\sqrt{1-\frac{b^2}{2^2}}\) Square both sides to eliminate the square root: \(\frac{1}{4} = 1 - \frac{b^2}{4}\) Now, solve for \(b^2\): \(\frac{b^2}{4} = 1 - \frac{1}{4}\) \(\frac{b^2}{4} = \frac{3}{4}\) \(b^2 = 3\)

Construct the equation of the ellipse

Now that we have the values of \(a^2\) and \(b^2\), we can construct the equation of the ellipse: \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) \(\frac{x^2}{2^2}+\frac{y^2}{3}=1\) The final equation of the ellipse becomes: \(4x^2 + 3y^2 = 12\) Comparing this to the given answer choices, we find that the correct option is: (c) \(4x^{2}+3y^2 = 12\)

Key concepts

Eccentricity

Eccentricity is a measure of how much an ellipse deviates from being a circle. It is denoted by the letter \(e\) and is calculated as the ratio between the distance from the center of the ellipse to one of its foci and the semi-major axis. In mathematical terms, the eccentricity \(e\) is defined as:\[e = \sqrt{1 - \frac{b^2}{a^2}}\]where \(a\) is the semi-major axis and \(b\) is the semi-minor axis.

For a circle, the eccentricity is always \(0\), since both axes are equal, while for an ellipse, it ranges between \(0\) and \(1\). A higher eccentricity indicates a more elongated shape. In this exercise, the given eccentricity is \(\frac{1}{2}\), suggesting a moderately elongated ellipse.

To determine the complete specifications of an ellipse, knowing the eccentricity is crucial as it works in conjunction with the semi-major and semi-minor axes and often with the directrix to give a full picture of the ellipse's shape and orientation.

Directrix

The directrix of an ellipse is a fixed straight line used in the formal definition of an ellipse. It is an important part of the construction that helps to define the shape along with the focus. When you know the eccentricity and the location of the directrix, you can further establish the relationship between the ellipse's other defining elements such as its axes.

In the given problem, one of the directrices is specified by the line \(x = 4\), implying that it is a vertical line parallel to the y-axis. This also means that our semi-major axis has to be along the x-direction, creating an alignment with the definition of an ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).

The directrix aids in constructing an ellipse by providing a reference for measuring distances from the focus and utilizing the eccentricity. It plays a crucial role in equations like \(\frac{a}{e} = \text{distance from center to directrix}\), as seen in the problem at hand.

Semi-major axis

The semi-major axis \(a\) of an ellipse is its longest radius. It extends from the center to the farthest point on the perimeter, passing through both foci of the ellipse. It plays a central role in the equation of the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where \(a > b\).

In this exercise, given the relationship \(\frac{a}{e} = 4\), we compute \(a = 2\) when the eccentricity \(e\) is \(\frac{1}{2}\). The semi-major axis is crucial because it determines both the scale and orientation of the ellipse. It impacts the overall shape and influences how elongated the ellipse is, relative to being a perfect circle (where \(a=b\)).

The semi-major axis combined with the eccentricity can help delineate other segments of the ellipse, like the location of the foci and vertices.

Semi-minor axis

The semi-minor axis \(b\) of an ellipse is its shortest radius. This line runs perpendicular to the semi-major axis and is crucial in shaping the ellipse. It helps determine the degree of flattening or roundedness of an ellipse.

Using the relationship for eccentricity given by \(e = \sqrt{1 - \frac{b^2}{a^2}}\), we find \(b^2 = 3\) in this problem. Solving this provides key information when plugging back into the ellipse equation. This radius plays a pivotal role in conjunction with the semi-major axis \(a\) for crafting the ellipse's equation and visualizing its form, \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).

The semi-minor axis affects the elliptical curvature and when altered, shifts the ratio of the axes, leading to variations in eccentricity, hence impacting the ellipse's overall geometry.