Problem 64 - A Complete Resource Book in Mathematics for JEE Main

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 tan gent, nearer to 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 the standard form is (A) \(\frac{\left(x-\frac{1}{3}\right)^{2}}{\frac{1}{9}}+\frac{(y-1)^{2}}{\frac{1}{12}}=1\) (B) \(\frac{\left(x-\frac{1}{3}\right)^{2}}{\frac{1}{12}}+\frac{(y-1)^{2}}{\frac{1}{9}}=1\) (C) \(\frac{(x-1)^{2}}{\frac{1}{9}}+\frac{\left(y-\frac{1}{3}\right)^{2}}{\frac{1}{12}}=1\) (D) none of these

Step-by-Step Solution

Answer

None of these choices match the calculated properties.

Step 1: Understanding Eccentricity

For an ellipse, the eccentricity \( e \) 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. We know \( e = \frac{1}{2} \). Therefore, \( c = \frac{a}{2} \).

Step 2: Understanding the Directrix

The directrix is a line such that distance from any point on the ellipse to a focus is \( e \) times the distance from the point to the directrix. As the given directrix is common and tangent to both the circle \( x^2 + y^2 = 1 \) and the hyperbola \( x^2 - y^2 = 1 \), we must identify where these curves meet.

Step 3: Calculating the Common Tangent

The common tangent to the circle and hyperbola is \( y = x \) and \( y = -x \). Note that the common tangent closer to the focus \( P(\frac{1}{2}, 1) \) is \( y = x \).

Step 4: Finding Distances to Apply Formula

Since the point \( P\left(\frac{1}{2}, 1\right) \) lies on the directrix \( y = x \), calculate the perpendicular distance from \( P \) to the directrix. This is zero, hence it provides no information directly. Instead, use \( c \) and \( a \) defined previously.

Step 5: Finalize the Equation

Checking the choices and the closest format, since \( a^2 = 4c^2 \) (using \( c = \frac{a}{2} \)), no matching option from the given matches the typical format for \( a^2 \), even with simplifications like shifting centers.

Key Concepts

Ellipse

An ellipse is one of the conic sections that is formed by cutting a cone with a plane at an angle that is different from the angle of the cone's surface. It is a symmetrical shape like a flattened circle. An ellipse has two axes: the major axis (the longest diameter) and the minor axis (the shortest diameter). The lengths of these axes determine the shape of the ellipse.

The general equation of an ellipse centered at the origin is given by:

• \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]

where \(a\) is the semi-major axis and \(b\) is the semi-minor axis. The shape of the ellipse depends on the ratio \( \frac{a}{b} \). When this ratio is close to 1, the ellipse looks more like a circle. Conversely, the greater the discrepancy between \(a\) and \(b\), the more elongated the ellipse becomes.

Eccentricity

Eccentricity is a measure that determines how "stretched" an ellipse is. For an ellipse, the eccentricity \( e \) is defined as the ratio \( e = \frac{c}{a} \), where \(c\) is the distance from the center to one of the foci, and \(a\) is the length of the semi-major axis.

Eccentricity can range between 0 and 1.

• If \( e = 0 \), the shape is a perfect circle because the foci coincide with the center.
• If \( e \) is close to 1, the ellipse is flatter.
• A larger \( e \) means a more elongated ellipse.

In the provided exercise, the eccentricity was given as \(\frac{1}{2}\), indicating a moderately elongated shape.

Directrix

A directrix of an ellipse is a line used to define and describe the curve in conjunction with the focus. For any point on the ellipse, the ratio of the distance to a focus and the distance to the directrix is constant, and is equal to the eccentricity, \( e \).

The concept of the directrix helps in constructing ellipses and understanding their geometric properties. For ellipses having very low eccentricity (i.e., closer to a circle), the directrix seems less impactful visually, but it mathematically defines the curve.

In the given exercise, the directrix is identified as a common tangent to the circle \(x^2+y^2=1\) and the hyperbola \(x^2-y^2=1\). The tangent line closer to the advertised focus point \(P\left(\frac{1}{2}, 1\right)\) was determined as \(y = x\).

Focus

An ellipse has two focal points, commonly referred to as the foci. These are located along the major axis, equidistant from the center of the ellipse. The role of the foci is crucial to the ellipse's definition: for any point on the ellipse, the sum of the distances to the two foci is constant, and equals the length of the major axis, \(2a\).

This means if you take any point on the edge of an ellipse and measure the distance to each focus, then add these two numbers together, they will always equal the same total. This property is what makes an ellipse unique from other conic sections.

In the specified problem, one focus is explicitly given: \(P\left(\frac{1}{2}, 1\right)\). Using knowledge of the eccentricity, the directrix, and other geometric properties, this allowed solving for the correct ellipse equation.