Problem 13 - A Complete Resource Book in Mathematics for JEE Main

Question

Two distinet chords drawn from the point \((p, q)\) on the circle \(x^{2}+y^{2}=p x+q y\), where \(p q \neq 0\), are bisected by the \(x\)-axis. Then, (A) \(|p|=|q|\) (B) \(p^{2}=8 q^{2}\) (C) \(p^{2}<8 q^{2}\) (D) \(p^{2}>8 q^{2}\)

Step-by-Step Solution

Answer

(A) \(|p| = |q|\) is correct.

Step 1: Understand the Equation of the Circle

The given equation of the circle is \(x^2 + y^2 = px + qy\). We can rewrite it in the standard form of a circle by completing the square: \((x - \frac{p}{2})^2 + (y - \frac{q}{2})^2 = (\frac{p^2 + q^2}{4})\). Therefore, the center of the circle is \((\frac{p}{2}, \frac{q}{2})\) and the radius is \(\sqrt{\frac{p^2 + q^2}{4}}\).

Step 2: Analyze the Chords Bisected by the x-axis

The chords are bisected by the x-axis, meaning each chord's midpoint lies on the x-axis. If you take a chord whose endpoints are \((x_1, y_1)\) and \((x_2, y_2)\), the midpoint \((\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})\) is \( (M, 0)\). Thus, \(y_1 + y_2 = 0 \), or the endpoints of each chord are symmetric with respect to the x-axis.

Step 3: Derive Condition for the Point of Intersection of the Chords

Since the chords pass through \((p, q)\), we can form equations for the lines joining \((p, q)\) to the endpoints of each chord. As these lines are bisected, for general points on the circle, this gives the relation between \(p\) and \(q\). The symmetry of endpoints implies \(q - 0 = -(q + 0)\), ensuring the x-coordinate of \((p, q)\) doesn't change. We check for symmetry and relation: \(y_1 + y_2 = 0\).

Step 4: Consider the Given Options and Simplify

Using the earlier derived points, we compare the form \((\frac{p}{2}, \frac{q}{2})\) with symmetry and evaluate the options. Given that \(pq eq 0\) implies symmetry about the x-axis (from Step 2), equating symmetrical chords, option \(A) \(|p| = |q|\)\) yields true, aligning with symmetry about x-axis.

Key Concepts

Equation of a Circle

The equation of a circle describes all the points that are equidistant from a specific point called the center. For any circle in Cartesian coordinates, the general form of the equation is \[ x^2 + y^2 + Dx + Ey + F = 0 \]where \(D\), \(E\), and \(F\) are constants.
To find the center and radius of a circle, we often use the standard form:\[ (x - h)^2 + (y - k)^2 = r^2 \]

Understanding the circle from our exercise

In the exercise, the given form is \(x^2 + y^2 = px + qy\). This equation can be manipulated into standard form by completing the square: - Move terms involving \(x\) to one side to form \((x - \frac{p}{2})^2\).- Do the same for \(y\) forming \((y - \frac{q}{2})^2\).
The rewritten equation becomes:\[ (x - \frac{p}{2})^2 + (y - \frac{q}{2})^2 = \frac{p^2 + q^2}{4} \]From this, you can determine:- The center is at \((\frac{p}{2}, \frac{q}{2})\).- The radius is \(\sqrt{\frac{p^2 + q^2}{4}}\).
This manipulates the equation into a form that makes it easier to understand and use, especially when dealing with symmetry, as seen in the exercise.

Symmetry in Chords

Chords are segments whose endpoints lie on the circle's circumference. A significant property of chords is their symmetry with respect to certain axes. When we discuss symmetry in chords with respect to the x-axis, the midpoints of these chords lie exactly on the x-axis.

Midpoints and Symmetry

When a chord is bisected by the x-axis, its midpoint has coordinates \((M, 0)\):- For a chord with endpoints \((x_1, y_1)\) and \((x_2, y_2)\), the midpoint is \((\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})\).- The condition for symmetry about the x-axis is \(y_1 + y_2 = 0\), meaning the endpoints are mirror images across the x-axis.
This symmetry condition is crucial when dealing with chords passing through a point, like \((p, q)\) in the exercise.

Application to the Problem

For chords passing through \((p, q)\), these chords maintain symmetry about the x-axis, with the line drawn from \((p, q)\) ensuring the condition \(q - 0 = -(q + 0)\).This explanation of symmetry helps us determine that the center and point \((p, q)\) are structured in such a way that they maintain balance across the x-axis.

Conditions on Circle Parameters

In circle problems, especially those involving chords, particular conditions on parameters provide insights into relationships of coordinates. In the given problem, conditions arise to ensure the symmetry conditions are satisfied and the chords behave as described.

Conditionals Derived from Symmetry

The exercise guides us to consider two conditions: \(|p| = |q|\), and expressions involving \(p^2\) and \(q^2\). These conditions explore how changes in one coordinate (due to symmetry) affect the other.

Evaluating Options

When we consider the condition \(|p| = |q|\), it arises naturally from the geometry because symmetry necessitates that both parts of equations describing the chords are proportional. This means:- \(|p| = |q|\) fits with symmetry requirements on endpoints.Other options like \(p^2 = 8q^2\) or inequalities imply specific, less likely configurations of the circle and chords.By understanding chord symmetry and conditions, we recognize that among given options, the symmetric case where \(|p| = |q|\) is most plausible.