Problem 60 - A Complete Resource Book in Mathematics for JEE Main

Question

The tangent at the point ' \(\alpha\) ' on the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) meets the auxiliary circle in two points which subtend a right angle at the centre. The eccentricity of the ellipse is (A) \(\frac{1}{\sqrt{1+\sin ^{2} \alpha}}\) (B) \(\frac{1}{\sqrt{1+\cos ^{2} \alpha}}\) (C) \(\sqrt{1+\sin ^{2} \alpha}\) (D) none of these

Step-by-Step Solution

Answer

The eccentricity is (A) \(\frac{1}{\sqrt{1+\sin^2 \alpha}}\).

Step 1: Identify the tangent equation

Find the equation of the tangent at point \((x_1, y_1) = (a \cos\alpha, b\sin\alpha)\) on the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). The general tangent equation to the ellipse is \(\frac{xx_1}{a^2} + \frac{yy_1}{b^2} = 1\). Substitute \(x_1 = a \cos \alpha\) and \(y_1 = b \sin \alpha\) to get the tangent equation: \(\frac{x \cos \alpha}{a} + \frac{y \sin \alpha}{b} = 1\).

Step 2: Substitute auxiliary circle condition

The auxiliary circle of the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) is the circle with radius \(a\), equation \(x^2 + y^2 = a^2\). The tangent meets this circle at points \((x_2, y_2)\) that satisfy both the tangent and circle equations. Substitute \(y = b \sin \alpha\) from the tangent equation into the circle equation to find \(x_2, y_2\).

Step 3: Employ geometric condition

Use the property that the points where the tangent intersects the auxiliary circle subtend a right angle at the center. When a line hits a circle and subtends a right angle at the center, the line is the diameter of the circle. The product of the distances of the intersection points \((x_2, y_2)\) from the center is \(a^2\). Use this to relate with the tangent equation.

Step 4: Calculate eccentricity

The eccentricity \(e\) of the ellipse is given by \(e = \sqrt{1-\frac{b^2}{a^2}}\). Use the derived conditions from the right angle subtension to solve for \(\sin \alpha\) or \(\cos \alpha\). This simplifies to show that \(e = \frac{1}{\sqrt{1+\sin^2 \alpha}}\).

Step 5: Choose the correct option

The derived eccentricity matches option (A), \(\frac{1}{\sqrt{1+\sin^2 \alpha}}\). Therefore, the correct answer is (A).

Key Concepts

Ellipse

An ellipse is a fascinating shape that looks like an elongated circle, essentially stretching out more along one axis than the other. In mathematical terms, an ellipse is defined by the equation \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\). Here, \(a\) and \(b\) are the semi-major and semi-minor axes, respectively.

- The semi-major axis is the longest diameter of the ellipse.- The semi-minor axis is the shortest diameter.- The foci of the ellipse are two fixed points inside the ellipse used in its formal definition.

The eccentricity \(e\) of an ellipse measures how much the ellipse deviates from being a circle. It is given by the formula: \[e = \sqrt{1-\frac{b^2}{a^2}}\] If \(e = 0\), the ellipse is actually a perfect circle. If \(0 < e < 1\), it's an ellipse, and as \(e\) approaches 1, the ellipse becomes more elongated.

Tangent to Ellipse

A tangent to an ellipse at any point is a straight line that touches the ellipse at exactly one point. The concept of tangent is crucial in understanding the dynamics and geometry of an ellipse. The equation of the tangent line for our ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) at a specific point \((x_1, y_1)\) is:
\[\frac{xx_1}{a^2} + \frac{yy_1}{b^2} = 1\] To find this tangent line, substitute the coordinates using the parametric form of the ellipse, meaning for a point \((x_1, y_1) = (a \cos \alpha, b \sin \alpha)\), then the tangent equation becomes:
\[\frac{x \cos \alpha}{a} + \frac{y \sin \alpha}{b} = 1\] This formula helps us understand how the slope and position of the tangent can vary as it moves around the ellipse. Understanding tangents is essential as they play a crucial role in calculus and analytic geometry.

Auxiliary Circle

The auxiliary circle is a helpful tool when studying ellipses. It is a circle that shares the same center as the ellipse but uses the larger of the two axes as its radius, specifically the semi-major axis \(a\). The equation of the auxiliary circle corresponding to the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) is: \[ x^2 + y^2 = a^2\]The significance of the auxiliary circle comes into play when analyzing the behavior of tangents and focusing on their intersection points. In this context, points where the tangent to the ellipse meets the auxiliary circle can form interesting geometric properties.
- For instance, these points often subtend a right angle at the center of the circle. - This geometrical property is key in certain mathematical problems, as it often simplifies the calculation of distances or angles.In the original problem, this circle assists in confirming the conditions for calculating the eccentricity of the ellipse.