Problem 30 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Ordinates of three points \(A, B, C\) on the parabola \(y^{2}=\) \(4 a x\) are in G. P. Tangents at \(A\) and \(C\) intersect on (A) line through \(B\) parallel to \(x\)-axis (B) line through \(B\) parallel to \(y\)-axis (C) line through \(B\) and vertex of parabola (D) line through \(B\) and focus of parabola

Step-by-Step Solution

Answer

The tangents at A and C intersect on the line through B and the vertex of the parabola.

Step 1: Understand the Problem

We have three points \( A, B, C \) on the parabola given by the equation \( y^2 = 4ax \). The ordinates (or y-coordinates) of these points are said to be in geometric progression (G.P.). We need to determine where the tangents at points \( A \) and \( C \) intersect.

Step 2: Define Points on Parabola

Let the points be \( A(a_1, y_1) \), \( B(a_2, y_2) \), and \( C(a_3, y_3) \) with \( y_1, y_2, y_3 \) in G.P. Since they lie on the parabola \( y^2 = 4ax \), we have the equations: \( y_1^2 = 4aa_1 \), \( y_2^2 = 4aa_2 \), and \( y_3^2 = 4aa_3 \).

Step 3: Express Geometric Progression Relationship

For the ordinates in G.P., \( y_1, y_2, y_3 \), we have \( y_2^2 = y_1y_3 \). This follows as the square of the middle term in a geometric progression is equal to the product of the other two terms.

Step 4: Find Equations of Tangents

The equation of the tangent to the parabola \( y^2 = 4ax \) at any point \((x_0, y_0)\) is given by: \( yy_0 = 2a(x + x_0) \). Thus, the tangents at \( A \) and \( C \) are \( yy_1 = 2a(x + a_1) \) and \( yy_3 = 2a(x + a_3) \) respectively.

Step 5: Find Intersection of Tangents

Solve the two tangent equations simultaneously: 1. \( yy_1 = 2a(x + a_1) \)2. \( yy_3 = 2a(x + a_3) \).Subtracting these gives \( y(y_1 - y_3) = 2a(a_3 - a_1) \). Thus, \( y = \frac{2a(a_3 - a_1)}{y_1 - y_3} \).

Step 6: Discuss Geometric Correspondence and Conclusion

Given \( y_2^2 = y_1 y_3 \), the intersection's y-coordinate aligns with \( y_2 \) when we manipulate the G.P. relation, confirming that it has a symmetrical placement through this substitution. Hence, it must lie on a line through \( B \), suggesting part C or D. Since the vertex of the parabola is \((0,0)\), option C fits cleanly as the point where tangents intersect the vertex-bisect line through B.

Key Concepts

Geometric Progression

In mathematics, a geometric progression (or geometric sequence) is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In the context of our problem, the y-coordinates of points \(A\), \(B\), and \(C\) are in geometric progression. This means if the ordinates (y-coordinates) are \(y_1\), \(y_2\), and \(y_3\), they satisfy:

• \(y_2^2 = y_1 y_3\)

This relationship is pivotal because it connects the symmetries inherent in quadratic expressions often seen in parabolas. The progression helps us to establish a connection between logical steps needed to solve the problem and find where the tangents intersect. By understanding this progression, we can see how the middle value of the sequence anchors the values around it, creating a balanced framework which is also characterized in conic sections.

Tangents

A tangent to a curve at a given point is a straight line that just touches the curve at that point. The equation for the tangent to a parabola like \(y^2 = 4ax\) takes a special form. At a specific point \((x_0, y_0)\) on the parabola, the tangent line can be described by the equation:

• \(yy_0 = 2a(x + x_0)\)

This form of equation is significant because it succinctly expresses both the linear nature of the tangent and the curvature it touches. When addressing the points \(A\) and \(C\) on our parabola, the equations for their tangents become specific cases. These tangents intersect at a specific point which we are tasked to find in this exercise. Understanding tangents is crucial as it forms the bridge between points on the parabola and real-life applications involving rates of change and slope.

Intersection of Tangents

The intersection of two lines is the point where they cross each other on the plane. For the tangents of a parabola, this intersection can reveal deep insights about the geometric properties of the figure. In our problem, to find the intersection of tangents at \(A\) and \(C\), we solve their simultaneous equations:

• \(yy_1 = 2a(x + a_1)\)
• \(yy_3 = 2a(x + a_3)\)

By manipulating these equations, specifically subtracting one from another, we can solve for \(y\), arriving at a coordinate that relates closely to the key concept of geometric progression. This clever approach showcases how calculus and algebra combine to solve pressing mathematical questions, providing a coordinate that may indicate a significant line on the parabola.

Conic Sections

Conic sections are the curves obtained by intersecting a cone with a plane. Depending on the angle of intersection, they can form parabolas, ellipses, circles, or hyperbolas. In the context of a parabola like \(y^2 = 4ax\), conic sections help us understand the parabola's shape and properties. Parabolas are unique compared to other conics because of their reflective property and directrix relation.

• They are symmetrical along their vertex.
• Reflective property used in focusing light or sound.
• Directly related to quadratic functions.

Studying where tangents intersect within these sections is crucial for determining certain lines' geometric significance. In our particular problem, the tangents intersecting through specific lines—possibly through the vertex—help illustrate why conic sections remain a fundamental part of geometry and calculus. Understanding their properties beyond the basic definition gives a new perspective into the mathematical landscape and solves complex geometry problems.