2010 IIT JEE Paper 1 Problem 50 Hyperbola Eccentricity | Summary and Q&A

TL;DR

The eccentricity of a hyperbola is determined by the ratio of the distance from any point on the hyperbola to its closest directrix to the distance from that point to its focus.

Key Insights

• 🫥 The equation of the tangent line is used to find the eccentricity of a hyperbola.
• 👈 The point of intersection of the tangent line and the x-axis represents one of the directrices.
• 😚 The distance from a point on the hyperbola to its closest directrix is related to the eccentricity.

Transcript

The line 2x plus y equals 1 is tangent to the hyperbola x squared over a squared minus y squared over b squared equals 1. If this line passes through the point of intersection of the nearest directrix and the x-axis, then the eccentricity of the hyperbola is-- so let's just set this up so we know what this is even asking. Let me draw the hyperbola ... Read More

Questions & Answers

Q: How is the eccentricity of a hyperbola defined?

The eccentricity of a hyperbola is the ratio of the distance from any point on the hyperbola to its closest directrix to the distance from that point to its focus.

Q: How is the equation of the tangent line used to find the eccentricity?

The equation of the tangent line is used to substitute for the slope and y-intercept in the formula relating the tangent line to the hyperbola. This equation, along with other known information about the hyperbola, allows for the determination of the eccentricity.

Q: How is the point of intersection of the tangent line with the x-axis used in finding the eccentricity?

The point of intersection of the tangent line with the x-axis represents the x-coordinate of one of the directrices. By using this point along with other known information about the hyperbola, the distance from the directrix to the focus can be determined, which is essential in finding the eccentricity.

Q: Why is the eccentricity of the hyperbola always greater than 1?

The eccentricity of a hyperbola is defined as the ratio of the distance from a point on the hyperbola to its closest directrix to the distance from that point to its focus. For a hyperbola, the focus is further from the point on the curve than the directrix, resulting in a ratio greater than 1.

Summary & Key Takeaways

• The given hyperbola has a tangent line of 2x + y = 1, which passes through the point of intersection of the nearest directrix and the x-axis.

• By setting y = 0, the point of intersection is (1/2, 0), and the directrix is x = 1/2.

• Using formulas for hyperbolas, tangents, and directrices, the eccentricity of the hyperbola can be determined.