Question

The length of latus rectum of the ellipse whose focus is \(\mathrm{S}(-1,1)\), the corresponding directrix is \(x-y+3=0\) and the eccentricity is \(\frac{1}{2}\), is equal to (1) \(\frac{1}{2}\) (2) \(\frac{1}{2 \sqrt{2}}\) (3) \(\frac{1}{\sqrt{2}}\) (4) \(\frac{1}{4}\)

Short answer

The length of the latus rectum is \( \frac{3}{2} \).

Step-by-step solution

- Identify and write the directrix equation

The directrix of the ellipse is given by the equation: \[ x - y + 3 = 0 \]

- Write the coordinates of the focus

The coordinates of the focus are given as: \[ S(-1, 1) \]

- Establish the relationship involving the eccentricity

The eccentricity (e) of the ellipse is given as: \[ e = \frac{1}{2} \]

- Calculate the perpendicular distance from the focus to the directrix

Using the formula for the distance from a point \((x_1, y_1)\) to the line \(Ax + By + C = 0\): \[ \text{distance} = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]Here, A = 1, B = -1, C = 3, and \((x_1, y_1) = (-1, 1)\), so the distance becomes: \[ \text{distance} = \frac{|1(-1) - 1(1) + 3|}{\sqrt{1^2 + (-1)^2}} = \frac{| -1 -1 + 3 |}{\sqrt{2}} = \frac{|1|}{\sqrt{2}} = \frac{1}{\sqrt{2}} \]

- Use the eccentricity to find the semi-major axis ‘a’

Recall the relationship between the focus, directrix, and eccentricity: \[ e = \frac{d}{a} \]where d is the distance from the focus to the directrix. Plugging in the values, we get: \[ \frac{1}{2} = \frac{ \frac{1}{\sqrt{2}} }{a} \]Solving for 'a', we find: \[ a = \frac{1}{\sqrt{2}} \times 2 = \sqrt{2} \]

- Determine the length of the latus rectum

The length of the latus rectum (L) of an ellipse is given by: \[ L = \frac{2b^2}{a} \]where \(b\) is the semi-minor axis. Using the relationship \( b = a \sqrt{1 - e^2} \) and given \( e = \frac{1}{2} \), we find:\[ b = \sqrt{2} \cdot \sqrt{ 1 - (\frac{1}{2})^2} = \sqrt{2} \cdot \sqrt{ \frac{3}{4} } = \sqrt{2} \cdot \frac{\sqrt{3}}{2} = \frac{ \sqrt{6} }{2} \]Now, the length of the latus rectum is:\[ L = \frac{2 ( \frac{\sqrt{6}}{2} )^2}{\sqrt{2}} = \frac{2 \cdot \frac{6}{4}}{\sqrt{2}} = \frac{3}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{3\sqrt{2}}{2} \times \frac{1}{\sqrt{2}} = \frac{3}{2} \]

Key concepts

Latus Rectum

The latus rectum of an ellipse is a fascinating aspect that relates directly to its geometric properties. It’s the line segment perpendicular to the major axis of the ellipse and passing through a focus. Importantly, it also intersects the ellipse. The length of the latus rectum is useful for various calculations. It’s given by the formula: \[ L = \frac{2b^2}{a} \] where \(b\) is the semi-minor axis and \(a\) is the semi-major axis. In our example, after calculating the values of \(a\) and \(b\), the length of the latus rectum of the given ellipse is found to be \( \frac{3}{2} \). This calculation emphasizes the interconnectedness of an ellipse’s parameters and how each element contributes to its overall shape. Understanding the latus rectum helps in visualizing the ellipse better and is pivotal in problems dealing with conic sections.

Eccentricity

Eccentricity, denoted as \(e\), is a measure of how much an ellipse deviates from being circular. It is a crucial parameter in defining the shape of the ellipse. Eccentricity ranges from 0 to 1 for ellipses. A circle is a special case of an ellipse with an eccentricity of 0. For our problem, the eccentricity is given as \( \frac{1}{2} \). This value indicates that the ellipse is reasonably closer to a circle but still noticeably 'stretched' along one axis. The formula to find the relationship involving eccentricity in an ellipse is: \[ e = \frac{c}{a} \] where \(c\) is the distance from the center to a focus, and \(a\) is the semi-major axis. Eccentricity helps in understanding the elongation of the ellipse and impacts other attributes, such as the length of the latus rectum. Remember, the smaller the eccentricity value, the rounder the ellipse.

Directrix

The directrix of an ellipse is one of the two fixed lines used to describe the locus of points that form the ellipse. In simpler terms, an ellipse can be defined as the set of all points where the ratio of the distance of each point to a focus and to the directrix is constant and equal to the eccentricity. The equation of a directrix is generally given in the form of a linear equation. In our case, the directrix is \[ x - y + 3 = 0 \] The relationship between the focus, directrix, and eccentricity is crucial in solving problems related to ellipses. By employing the distance formula, we utilized the directrix to find the semi-major axis, which further allowed us to compute various other components such as the latus rectum. Understanding the role of the directrix offers deeper insights into the geometrical layout of ellipses and is fundamental in conic sections.