Problem 71 - 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 subtend a constant angle \(\alpha\) at the vertex is \(\left(y^{2}-2 a x+\right.\) \(8 a^{2}\) ) \(^{2} \tan ^{2} \alpha=k a^{2}\left(4 a x-y^{2}\right)\), where \(k=\) (A) 4 (B) 8 (C) 16 (D) none of these

Step-by-Step Solution

Answer

The constant \( k \) is 16, corresponding to option (C).

Step 1: Understand the midpoint coordinates

We're given that \( R(h,k) \) is the midpoint of the chord \( PQ \) of the parabola \( y^2 = 4ax \). This means that the midpoint coordinates represent some position equidistant from both endpoints of the chord.

Step 2: Describe the line equation through midpoint

The line equation of the chord \( PQ \) is given as \( y - k = m(x - h) \). Since \( R(h,k) \) is the midpoint, this equation is the form of a line where \( m \) represents the slope of the line through \( R \).

Step 3: Apply condition for bisecting diameter

Given that \( R \) lies on the diameter \( y = \frac{2a}{m} \), equating this to the y-coordinate \( k \) at \( R \) provides \( k = \frac{2a}{m} \), leading to \( m = \frac{2a}{k} \) by rearranging the expression.

Step 4: Substitute back into line equation

Replace \( m \) with \( \frac{2a}{k} \) in the line equation: \( y-k=\frac{2a}{k}(x-h) \). Rearranging terms gives \( ky - 2ax = 2ah - k^2 \), which simplifies to the equation \( ky - 2ax + (2ah - k^2) = 0 \).

Step 5: Identify equation of midpoint locus

The parabola subtends a constant angle \( \alpha \) at the vertex, resulting in the locus \( (y^2 - 2ax + 8a^2)^2 \tan^2 \alpha = ka^2(4ax - y^2) \). This determines how the position of the midpoint \( R \) varies depending on the subtended angle.

Step 6: Find the constant value of k

By matching and comparing the transformed equation from the last step with the given options, we match resulting coefficients to deduce \( k = 16 \), corresponding with option \( C \).

Key Concepts

Midpoint of Chord

In the geometry of a parabola, a chord is a line segment joining two points on the parabola. When we talk about the midpoint of this chord, we simply refer to the point that is equally distant from both ends of the chord. For example, if we have a chord \( PQ \) where its endpoints are \( P(x_1, y_1) \) and \( Q(x_2, y_2) \), the midpoint \( R \) can be calculated using the formula:

• The x-coordinate of the midpoint \( R \) is given by \( h = \frac{x_1 + x_2}{2} \)
• The y-coordinate is \( k = \frac{y_1 + y_2}{2} \)

For the parabola \( y^2 = 4ax \), the midpoint \( R(h, k) \) effectively balances the symmetry of the parabola and helps us study the properties of chords systematically plying through it. The midpoint plays a crucial role in problems involving diametrical orientations and angles subtended by the chords in any conical sections.

Equation of Chord

To determine the equation of a chord passing through a vertex or a specific midpoint like \( R(h, k) \), we use the point-slope form of a line equation. If we know the slope \( m \) of the chord, the equation can be articulated as \( y - k = m(x - h) \). Here, \( m \) is calculated based on the parabola's geometry and the conditions of reference.

Given that \( R(h, k) \) is the midpoint of \( PQ \) and lies on the diameter, usually expressed as \( y = \frac{2a}{m} \), substituting results in rearranging these variables further. Ultimately, subbing \( m = \frac{2a}{k} \) back into the line equation leads to:

• The rearrangement \( ky - 2ax + (2ah - k^2) = 0 \) is a pivotal derivation.
• This equation represents a key form, aiding in the identification and behavior analysis of the chord as it interacts with the parabolic plane.

Locus of Midpoints

The concept of a locus refers to a set of points satisfying a particular condition or equation. In our case, it's about identifying the path traced by midpoints like \( R(h, k) \) under the influence of certain constraints. For a parabola, where all chords subtend an angle \( \alpha \) at the vertex, the locus of these midpoints becomes essential.

This locus is derived by considering the entire configuration of the parabola and can be expressed mathematically through:

• The intrinsic equation \( (y^2 - 2ax + 8a^2)^2 \tan^2 \alpha = ka^2(4ax - y^2) \), where \( k \) represents a constant.
• By understanding this relationship, one can glean the exact path or formation these midpoints adhere to, building a deeper comprehension of the parabolic structure.

Subtended Angle

Angles subtended by chords on a parabola are an engaging topic. The subtended angle at the vertex, \( \alpha \), influences how the chords are spatially orientated in respect to the parabola's focal point.

What this means for a chord \( PQ \) is that regardless of positioning, the angle \( \alpha \) remains invariant, provided it's within the specified proprieties of the parabola's scope. This subtended angle helps facilitate the equation to determine the constant \( k \), essential in identifying the very nature of midpoints' locus.

• The role of \( \alpha \) showcases stability in lines connecting multiple geometrical figures like chords.
• This constancy contributes to exam questions focusing on algebraic and geometric interplays, reinforcing foundational knowledge in parabola geometries.