Problem 72 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(R(h, k)\) be the middle point of the chord \(P Q\) of the parabola \(y^{2}=4 a x\). Equation of \(P Q\) is $$ (y-k)=m(x-h) $$ where \(m\) is the slope of \(P Q\) \(\because R\) lies on the diameter \(y=\frac{2 a}{m}\) bisecting \(P Q\), $$ \therefore k=\frac{2 a}{m} \Leftrightarrow m=\frac{2 a}{k} $$ Subsituting this value of \(m\) in (1), we have \(y-k=\frac{2 a}{k}(x-h)\) or, \(k(y-k)=2 a(x-h)\) or \(k y-2 a x+\left(2 a h-k^{2}\right)=0\) which is the required equation. The locus of the middle point of chords of the parabola which passes through the focus is (A) \(y^{2}=a(x-a)\) (B) \(y^{2}=2 a(x-a)\) (C) \(y^{2}=4 a(x-a)\) (D) none of these

Step-by-Step Solution

Answer

The locus of the middle point is \(y^2 = 2a(x-a)\) (Option B).

Step 1: Find the Middle Point of the Chord

Given the parabola equation is \(y^2 = 4ax\). Let \(R(h, k)\) be the middle point of the chord \(PQ\). The slope \(m\) of the chord \(PQ\) is given, so the equation of \(PQ\) can be written as \((y-k) = m(x-h)\).

Step 2: Determine Diameter

Since \(R\) lies on the diameter \(y = \frac{2a}{m}\) bisecting \(PQ\), the relationship \(k = \frac{2a}{m}\) can be derived, leading to \(m = \frac{2a}{k}\).

Step 3: Substitute Slope into Equation

Substitute \(m = \frac{2a}{k}\) into the equation \(y - k = m(x - h)\), resulting in \(y - k = \frac{2a}{k}(x - h)\). Simplifying gives \(k(y-k) = 2a(x-h)\).

Step 4: Simplify the Equation

Rewrite the equation as \(ky - 2ax = 2ah - k^2\). Further simplification leads to \(ky - 2ax + (2ah - k^2) = 0\), which represents the general equation of the chord.

Step 5: Determine the Locus Passing Through the Focus

To find the locus of the middle point of a chord passing through the focus, substitute the focus \((a, 0)\) into the general chord equation \(ky - 2ax + (2ah - k^2) = 0\).

Step 6: Substitute Focus and Simplify

When substituting \((a, 0)\) into our equation, we get \(k(0) - 2a(a) + (2ah - k^2) = 0\), simplifying to \(-2a^2 + 2ah - k^2 = 0\). Solving for \(k^2\), you get \(k^2 = 2a^2 - 2ah\).

Step 7: Derive Locus Equation

Rearrange \(k^2 = 2a^2 - 2ah\) to express \(y^2\) in terms of \(x\). Since \(k = y\), replace \(k^2\) with \(y^2\) which gives \(y^2 = 2a(a-x) = 2a(x-a)\), matching option \(B\) of the problem.

Key Concepts

Chord

A chord is a line segment with both endpoints on the curve. In the context of a parabola, like the one expressed by the equation \(y^2 = 4ax\), a chord can be drawn from any two points \(P\) and \(Q\) on the parabola. The middle point of this chord, denoted as \(R(h, k)\), is significant in analyzing the parabola’s properties. When you know the slope \(m\) of the chord, you can write its equation as \((y - k) = m(x - h)\). This form is the point-slope form of a linear equation and is essential in analyzing how the chord interacts with the parabola. By understanding the concept of chords, students can explore how they interplay with geometrical features of the parabola.

Focus

The focus of a parabola is a fixed point that, along with the directrix (a fixed line), helps define the curve. For a parabola described by the equation \(y^2 = 4ax\), the focus is located at \((a, 0)\). This point is crucial in determining the unique shape and direction of the parabola. It acts as an arbitrary point from which the points on the parabola maintain equidistant relationships to the directrix. While dealing with chords that pass through the focus, you recognize that they have special properties, such as the fact that the equation of the locus of the middle points of these chords leads you to specific quadratic forms. Understanding the role of the focus aids in visualizing and characterizing the behaviour of parabolic curves.

Diameter

A diameter in the context of a parabola like \(y^2 = 4ax\), refers to a line segment that passes through the midpoint of a chord and is parallel to the axis of symmetry. In this problem, the diameter is given by the line \(y = \frac{2a}{m}\), where \(m\) is the slope of the chord. The diameter bisects the chord, making the midpoint \(R(h, k)\) lie on this line. This relationship, coupled with the perpendicular nature of the diameter and the chord, manifests in the equation \(k = \frac{2a}{m}\). Understanding diameters in the context of parabolas lets you see how chords are situated relative to the axis of symmetry and aids in exploring deeper geometric relationships in the parabola.

Slope

The slope is a measure of the steepness and direction of a line. For the line segment that is the chord \(PQ\), the slope is denoted by \(m\). In the equation \(m = \frac{2a}{k}\), the slope relates the vertical and horizontal distances between any two points on the line. By substituting \(m = \frac{2a}{k}\) into the equation of the chord, you begin to understand the dynamic interaction between the parabola’s curvature and the linearity of the chord. The slope tells you how much \(y\) changes for a unit change in \(x\), and it's crucial for drawing connections between linear equations and parabolic geometry. Mastering the concept of slope helps students appreciate a zest of calculus, embodying the linear approximation of curves.