Problem 35 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The line \(A x+B y+C=0\) cuts the circle \(x^{2}+y^{2}+a x+\) by \(+c=0\) in \(P\) and \(Q\). The line \(A^{\prime} x+B^{\prime} y+C^{\prime}=0\) cuts the circle \(x^{2}+y^{2}+a^{\prime} x+b^{\prime} y+c^{\prime}=0\) in \(R\) and \(S .\) If \(P, Q\), \(R, S\) are concyclic points, then (A) \(\left|\begin{array}{ccc}a+a^{\prime} & b+b^{\prime} & c+c^{\prime} \\ A & B & C \\ A^{\prime} & B^{\prime} & C^{\prime}\end{array}\right|=0\) (B) \(\left|\begin{array}{ccc}a-a^{\prime} & b-b^{\prime} & c-c^{\prime} \\ A & B & C \\ A^{\prime} & B^{\prime} & C^{\prime}\end{array}\right|=0\) (C) \(\left|\begin{array}{ccc}A\left(a+a^{\prime}\right) & B\left(b+b^{\prime}\right) & C\left(c+c^{\prime}\right) \\ A & B & C \\\ A^{\prime} & B^{\prime} & C^{\prime}\end{array}\right|=0\) (D) none of these

Step-by-Step Solution

Answer

Option (A) is correct.

Step 1: Understand the Problem

We have two circles and two lines. The first line intersects the first circle at points \( P \) and \( Q \). The second line intersects the second circle at points \( R \) and \( S \). We need to determine the condition under which these four intersection points \( P, Q, R, \) and \( S \) are concyclic (lie on the same circle).

Step 2: Condition for Concyclic Points

The points \( P, Q, R, \) and \( S \) are concyclic if the determinant formed by the equations of the circles and lines is zero. This determinant condition typically arises when considering the radical axis and common chords of the intersecting lines and circles.

Step 3: Apply the Determinant Condition

To determine concyclic points we use the determinant: \[\left|\begin{array}{ccc}a+a^{\prime} & b+b^{\prime} & c+c^{\prime} \ A & B & C \ A^{\prime} & B^{\prime} & C^{\prime}\end{array}\right| = 0\]This determinant ensures the right condition between the coefficients when the points are concyclic since it's based on their properties of intersection.

Step 4: Verification

Verify each option against the derived determinant condition. Evaluate each expression to see which one results in zero given the condition for concyclic points discovered in Step 3.

Key Concepts

Determinant Condition

Understanding the determinant condition is essential when solving problems involving concyclic points. In this particular problem, we have two lines that intersect two separate circles at four points: \(P, Q, R, \text{ and } S\). For these points to be concyclic, they must lie on the same circle. The determinant condition provides a method to verify such a condition by using the coefficients of the equations for both the circles and the lines.

Mathematically, points \( P, Q, R, \text{ and } S \) are concyclic if the determinant formed by specific combinations of these coefficients equals zero:

• The coefficients from the circle equations: \(a+a', b+b', c+c'\)
• The coefficients from the line equations: \( A, B, C \text{ and } A', B', C' \)

Evaluating the determinant:\[\left|\begin{array}{ccc}a+a' & b+b' & c+c' \A & B & C \A' & B' & C'\end{array}\right| = 0\]ensures that the four points are concyclic under the given conditions.

This determinant condition is not just a theoretical concept; it is often used to solve complex geometric problems where this particular relationship helps identify points that share the same circle.

Intersection of Line and Circle

The intersection of a line and a circle is a fundamental concept in geometry. It plays a crucial role in problems involving concyclic points, like the current one focuses on. When a line intersects a circle, it typically does so at two distinct points. These intersection points become crucial in determining further geometric properties, such as concyclicity.

In the given exercise, the line \(Ax + By + C = 0\) intersects the circle \(x^2 + y^2 + ax + by + c = 0\) at points \(P\) and \(Q\). Similarly, another line \(A'x + B'y + C' = 0\) intersects a different circle at points \(R\) and \(S\). These intersections help in determining whether the four points are concyclic.

• Finding the points \(P, Q, R, \text{ and } S\) involves solving the system of equations representing the line and circle.
• These solutions are critical as they form the basis of applying the determinant condition needed for verifying concyclicity.

When we solve the equations derived from the intersection of lines and circles, we gain the coordinates of the intersection points, which can then be analyzed for further geometric properties.

Concyclic Condition Verification

Verification of concyclicity is a way of substantiating whether four points truly lie on a single circle. In our problem, after arriving at the determinant condition derived from the intersection coefficients, we need to ensure that this condition holds true.

To verify the concyclic condition, evaluate the determinant:\[\left|\begin{array}{ccc}a+a' & b+b' & c+c' \A & B & C \A' & B' & C'\end{array}\right|\]If the determinant is equal to zero, the condition is met, implying that \(P, Q, R, \text{ and } S\) are indeed concyclic.

• Each term in the determinant represents coefficients from the equations of circles and lines.
• By checking if the determinant equals zero, we verify that all necessary geometric conditions for concyclicity hold.

This verification process ties together our exploration of interdependent geometric configurations, highlighting how algebra and geometry intertwine to explain properties such as being concyclic.