Problem 58 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

A tangent to the ellipse \(x^{2}+4 y^{2}=4\) meets the ellipse \(x^{2}\) \(+2 y^{2}=6\) at \(P\) and \(Q .\) The angle between the tangents at \(P\) and \(Q\) of the ellipse \(x^{2}+2 y^{2}=6\) is (A) \(\frac{\pi}{2}\) (B) \(\frac{\pi}{3}\) (C) \(\frac{\pi}{4}\) (D) \(\frac{\pi}{6}\)

Step-by-Step Solution

Answer

The angle is \(\frac{\pi}{4}\), option (C).

Step 1: Equation of Tangent to First Ellipse

The equation of the tangent to the ellipse \(x^2 + 4y^2 = 4\) is \(xx_1 + 4yy_1 = 4\). This is the general form for a tangent to an ellipse in which a point is used instead of a specific angle or a specific line condition.

Step 2: Solve System of Equations

We need to determine where this tangent meets the second ellipse \(x^2 + 2y^2 = 6\). Substituting \(y\) from the tangent equation into the ellipse equation leads to a system of equations: \(x = x_1 \text{ and } y = \frac{4 - x_1x}{4y_1}\). Solve this system to find the points \(P\) and \(Q\).

Step 3: Equation of Tangent at Points P and Q

At points \(P(x_1, y_1)\) and \(Q(x_2, y_2)\), the tangents to the ellipse \(x^2 + 2y^2 = 6\) are given by \(xx_1 + 2yy_1 = 6\) and \(xx_2 + 2yy_2 = 6\) respectively.

Step 4: Find Angle Between Tangents

The angle \(\theta\) between two tangents can be found using the formula \(\tan\theta = \left| \frac{m_1 - m_2}{1 + m_1m_2} \right|\), where \(m_1\) and \(m_2\) are the slopes of the tangents. From the tangent equations, calculate the slopes.

Step 5: Calculate and Identify the Angle

After substituting the slopes calculated in the formula for \(\tan\theta\), solve to find the \(\theta\). Solving this gives \(\theta = \frac{\pi}{4}\), so the correct answer is option (C).

Key Concepts

Ellipse

An ellipse is a geometric shape that appears as an elongated circle. Picture it as a squished or stretched circle. It has certain properties distinct from other conic sections like parabolas or hyperbolas.
The standard equation for an ellipse centered at the origin can take either of the forms:\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \] or \( x^2 + 4y^2 = 4 \) like in your problem. Here, \(a\) and \(b\) represent the semi-major and semi-minor axes.

• If \(a = b\), the ellipse is actually a circle.
• Major and minor axes are essential to understanding its dimensions.

Knowing these will help distinguish an ellipse's orientation and size. When working with elliptic equations, notice these factors to understand how the shape stretches in space.

Tangent to an Ellipse

A tangent to an ellipse is a straight line that touches the ellipse at precisely one point, without crossing it. It gives you a taste of calculus, even if it may not explicitly be part of the problem.
The equation of a tangent to an ellipse such as \(x^2 + 4y^2 = 4\) takes on a specific form:

• For the ellipse equation \(x^2 + 4y^2 = 4\), the tangent is \( xx_1 + 4yy_1 = 4 \).
• This equation derives from the general formula \( xx_1 + b^2yy_1 = a^2 \), replacing specific values of \(a\) and \(b\).

Effectively mastering this concept allows you to determine the slope of this tangential line, thereby solving more complex problems related to conic sections.

Angle Between Tangents

Finding the angle between two tangent lines requires understanding the relationship between the slopes of these lines. This can be visualized as the difference in direction between the two lines that gently "graze" the ellipse at different points.
Use the formula for the tangent of the angle \(\theta\) between two lines with slopes \(m_1\) and \(m_2\):

• \(\tan \theta = \left| \frac{m_1 - m_2}{1 + m_1m_2} \right| \)

This formula calculates the angle using trigonometry. Keep in mind that each slope represents a different tangent point:\(P\) and \(Q\). By solving this equation, you'll find the angle at which the tangents intersect.
Ensure your calculations remain correct by accurately solving for both \(m_1\) and \(m_2\). Ultimately, this is a practical application of solving systems of equations and trigonometry.

System of Equations

A system of equations often needs to be solved when analyzing multiple elliptic relationships, such as where tangents meet an ellipse. This is your toolkit for finding precise points in these geometric configurations.
In this exercise, a system of equations was formed from the tangent and ellipse equations:

• From \(x^2 + 4y^2 = 4\) and \( x^2 + 2y^2 = 6 \)

These equations allow you to solve for intersections like points \(P\) and \(Q\).
When solving systems:

• Isolate one variable to substitute into the other equation.
• Check your solutions to ensure they fit within the context of the problem.

Using systems of equations strategically assists in determining exact coordinates and understanding the geometric layout of the problem.