Problem 78 - A Complete Resource Book in Mathematics for JEE Main

Question

In the following questions an Assertion \((A)\) is given followed by a Reason (R). Mark your responses from the following options: (A) Assertion(A) is True and Reason(R) is True; Reason( \(\mathrm{R}\) ) is a correct explanation for Assertion(A) (B) Assertion(A) is True, Reason(R) is True; Reason(R) is not a correct explanation for Assertion(A) (C) Assertion(A) is True, Reason(R) is False (D) Assertion(A) is False, Reason(R) is True Assertion:The locus of the centre of the circle described on any focal chord of a parabola \(y^{2}=4 a x\) as diameter is \(y^{2}\) \(=2 a(x-a)\) Reason: If \(A\left(a t_{1}^{2}, 2 a t_{1}\right)\) and \(B\left(a t_{2}^{2}, 2 a t_{2}\right)\) be the extremities of a focal chord for the parabola \(y^{2}=4 a x\), then \(t_{12}=t_{2}=-1\)

Step-by-Step Solution

Answer

(A) Assertion(A) is True and Reason(R) is True; Reason(R) is a correct explanation for Assertion(A)

Step 1: Understanding the Problem

We need to evaluate the truth of the assertion and reason given, and determine if the reason correctly explains the assertion. The problem involves a parabola equation and properties related to circle diameters and focal chords.

Step 2: Analyzing the Assertion (A)

The assertion states that the locus of the center of the circle on any focal chord of the parabola \(y^2 = 4ax\) as diameter is \(y^2 = 2a(x-a)\). This involves using the properties of focal chords and the midpoint formula to establish the locus equation.

Step 3: Verifying the Assertion

The coordinates of the extremities of a focal chord are \(A(at_1^2, 2at_1)\) and \(B(at_2^2, 2at_2)\), with the condition \(t_1 t_2 = -1\). The midpoint, or center of the circle, is then \((\frac{a(t_1^2 + t_2^2)}{2}, a(t_1 + t_2))\). Solving the system with \(t_1 t_2 = -1\) leads us to the correct locus equation, \(y^2 = 2a(x-a)\). Thus, the assertion is true.

Step 4: Analyzing the Reason (R)

The reason claims that for a parabola \(y^2 = 4ax\), the focal chord extremities \((at_1^2, 2at_1)\) and \((at_2^2, 2at_2)\) have the property \(t_1t_2 = -1\). This is correct because for any focal chord of a parabola, the product of the parameters \(t_1\) and \(t_2\) is indeed \(-1\). Hence, the reason is true.

Step 5: Checking if Reason Explains Assertion

The reason provided directly relates to the assertion since it offers the necessary condition to verify the locus result. The condition \(t_1 t_2 = -1\) is crucial in deriving the midpoint (center) equation leading to the correct locus, connecting the reason precisely to the assertion's conclusion.

Key Concepts

Locus

In mathematics, the term "locus" refers to the set of all points satisfying particular conditions or equations. It is basically a way to describe the path or region traced out by these points. For example, in the case of the parabola given as \( y^2 = 4ax \), determining the locus of centers of circles constructed on focal chords involves finding all possible positions where these centers can exist as the chords vary.

• Here, the centers, which are midpoints of focal chords, trace a path that forms another curve.
• In this specific exercise, the locus is the parabolic equation \( y^2 = 2a(x-a) \).
• This means that as we draw circles on focal chords of the given parabola, their centers will fit this equation, creating a new path described by this new parabola.

Focal Chord

A focal chord in the context of a parabola is a line segment that passes through the focus of the parabola and has both of its endpoints lying on the parabola. The parabola we are dealing with, \( y^2 = 4ax \), provides a special situation among conic sections due to its relationship with focal chords. The extremities of a focal chord are often given by parameters \( A(at_1^2, 2at_1) \) and \( B(at_2^2, 2at_2) \).

• Importantly, these parameters follow the condition \( t_1 t_2 = -1 \), meaning their product equals negative one.
• This condition is essential because it directly ties into the parabolic properties, allowing further calculations, such as finding midpoints or determining the locus of points associated with the parabola.

Understanding focal chords is crucial for grasping more complex geometry problems involving parabolas and their properties.

Midpoint Formula

The midpoint formula is a fundamental concept in geometry that helps us find the center of a line segment, and it is vital in coordinate geometry tasks. To calculate the midpoint of a line segment with endpoints \( (x_1, y_1) \) and \( (x_2, y_2) \), you use the equation: \[ \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]. This formula plays a key role in the given exercise:

• By using this method, we can determine the center of the circles that we create using focal chords as the diameter.
• The midpoint formula is applied directly to the coordinates \( A(at_1^2, 2at_1) \) and \( B(at_2^2, 2at_2) \).
• The center or midpoint ends up at \( \left( \frac{a(t_1^2 + t_2^2)}{2}, a(t_1 + t_2) \right) \).

Through this midpoint finding approach, we establish the locus equation \( y^2 = 2a(x-a) \), which informs us about how centers of these unique circles behave.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. It uses algebraic principles to address geometric problems. This fusion of algebra and geometry lets us handle complex scenarios with ease, like understanding parabolas, ellipses, and other figures by using equations. For a parabola such as \( y^2 = 4ax \), coordinate geometry not only helps plot these curves but also delve deeper into properties such as:

• Locating key points like the focus, vertex, and creating lines such as tangents and chords.
• Handling complicated computations regarding the relationships between various points on the curve using simple algebraic solutions.
• Transforming geometric conditions into algebraic expressions that describe loci or solutions.

In this exercise, applying coordinate geometry techniques allows us to analyze focal chords, employ the midpoint formula, and find the locus of the center points of circles, thus providing a comprehensive view of the problem statement.