Problem 20 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

An ellipse has eccentricity \(\frac{1}{2}\) and one focus at the point \(P\left(\frac{1}{2}, 1\right) .\) Its one directrix is the common tangent at the point \(P\), to the circle \(x^{2}+y^{2}=1\) and the hyperbola \(x^{2}-y^{2}=1 .\) The equation of the ellipse in standard form is (A) \(9\left(x-\frac{1}{3}\right)^{2}+(y-1)^{2}=1\) (B) \(9\left(x-\frac{1}{3}\right)^{2}+12(y-1)^{2}=1\) (C) \(\frac{\left(x-\frac{1}{3}\right)^{2}}{4}+\frac{(y-1)^{2}}{3}=1\) (D) none of these

Step-by-Step Solution

Answer

The correct option is (D) none of these.

Step 1: Understanding Eccentricity

The eccentricity of an ellipse is given by \( e = \frac{c}{a} \), where \( c \) is the distance from the center to a focus, and \( a \) is the semi-major axis length. Here, \( e = \frac{1}{2} \). Since we know one focus is at \( P\left(\frac{1}{2}, 1\right) \), the center of the ellipse must be \( \left(0, 1 \right) \). Since the center and focus are aligned vertically, the semi-major axis is horizontal.

Step 2: Finding the Lengths of the Axes

We know \( e = \frac{1}{2} \), so \( c = \frac{a}{2} \). The distance \( c \) is \( 0.5 \), thus \( a = 1 \) (since half of \( a \) should equal \( 0.5 \)). The ellipse's equation is of the form \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \). Since \( a = 1 \), \( a^2 = 1 \), and we need to find \( b^2 \).

Key Concepts

Eccentricity

The **eccentricity** of a conic section is a measure of how much it deviates from being a circle. For ellipses, it is represented by the letter \(e\). The eccentricity of a circle is zero, while for an ellipse, it is between zero and one. This value helps determine the shape of the ellipse, being less eccentric for values closer to zero and more elongated for values closer to one.

• In the context of the given problem, the eccentricity is \(\frac{1}{2}\), meaning the ellipse is fairly round but not as perfect a circle
• The formula for eccentricity of an ellipse is given by \(e = \frac{c}{a}\), where \(c\) is the distance from the center to a focus, and \(a\) is the length of the semi-major axis.
• For our problem, with \(e = \frac{1}{2}\), if the focus \(P\left(\frac{1}{2}, 1\right)\) is known, it means the distance from the center to the focus is half of the semi-major axis's length.

Ellipse Equation

The **ellipse equation** in standard form provides a way to describe an ellipse algebraically. It takes the form: \[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \] where \( (h, k) \) are the coordinates of the center of the ellipse, \( a \) is the semi-major axis, and \( b \) is the semi-minor axis.

• In the given problem, the center of the ellipse is at \( (0, 1) \).
• We found \(a = 1\), hence \(a^2 = 1\).
• The next step is determining \( b^2 \) by relation \( b^2 = a^2(1 - e^2)\), considering \( e = \frac{1}{2} \).

From the eccentricity, \( c = \frac{a}{2} = 0.5\), and using the formula for the relationship between \(a\), \(b\), and \(c\): \[ b^2 = a^2 - c^2 \] Thus, \( b^2 = 1 - 0.25 = 0.75\), leading to further clarification of the ellipse's features.

Conic Sections

**Conic sections** are the curves obtained from intersecting a plane with a cone, leading to various interesting shapes: circles, ellipses, parabolas, and hyperbolas. These shapes have unique properties and equations that govern them.

• A circle is a special case of an ellipse where both axes are equal, leading to an eccentricity of zero.
• An ellipse, on the other hand, features different axis lengths, thus possessing an eccentricity between zero and one.
• A parabola has only one directrix and an eccentricity of one.
• A hyperbola has an eccentricity greater than one, portraying distinct shapes with two branches.

In the context of many conic sections, understanding the eccentricity and directrix helps underline the relationships between focus and directrix, which defines these sections vividly. The problem ties directly into the elegant intersection of several geometric constructs (ellipse, circle, and hyperbola) at a particular point, highlighting the interconnectedness of various conic sections.