Problem 81 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If the circles \(x^{2}+y^{2}+2 a x+c y+a=0\) and \(x^{2}+y^{2}-3 a x+d y-1=0\) intersect in two distinct points \(P\) and \(Q\) then the line \(5 x+b y-a=0\) passes through \(P\) and \(Q\) for (a) exactly one value of \(a\) (b) no value of \(a\) (c) infinitely many values of \(a\) (d) exactly two values of \(a\)

Step-by-Step Solution

Answer

The line passes through P and Q for infinitely many values of a (option c).

Step 1: Understand the Condition for Circle Intersection

For two circles to intersect at two distinct points, the distance between the centers of the circles must be less than the sum of their radii and greater than the absolute value of the difference of their radii.

Step 2: Identify the Centers and Radii of the Circles

The general form of a circle is \(x^2 + y^2 + 2gx + 2fy + c = 0\). Rewriting the given equations, the first circle has center \((-a, -\frac{c}{2})\) and radius \(\sqrt{a^2 + \left(\frac{c}{2}\right)^2 - a}\). The second circle has center \((\frac{3a}{2}, -\frac{d}{2})\) and radius \(\sqrt{\left(\frac{3a}{2}\right)^2 + \left(-\frac{d}{2}\right)^2 - 1}\).

Step 3: Calculate Distance Between Centers

The distance \(D\) between the centers \((-a, -\frac{c}{2})\) and \((\frac{3a}{2}, -\frac{d}{2})\) is given by \[ D = \sqrt{\left(-a - \frac{3a}{2}\right)^2 + \left(-\frac{c}{2} + \frac{d}{2}\right)^2}. \]

Step 4: Use Intersection Condition

Set up the inequalities for the circles to intersect in two points: 1. \(D < \text{radii sum} \) 2. \(D > \text{radii absolute difference} \). Solve these inequalities with the expressions from Step 2 and Step 3.

Step 5: Conclude Based on Simplification

Simplify the inequalities. Check if any specific condition on \(a\) comes from comparing centers and solving the inequalities. You'll find that satisfying the condition leads to infinitely many solutions for \(a\) since intersections depend on conditions not reducing \(a\) to a constraint.

Key Concepts

Centers of Circles

Understanding the centers of circles is key when analyzing how they interact and intersect with one another. The center of a circle, in a 2D coordinate system, is the point from which all the points on the circle are equidistant. In the equation of a circle given by the general form \(x^2 + y^2 + 2gx + 2fy + c = 0\), the center is \((-g, -f)\).

For the circles in the problem, you need to identify their centers from their algebraic equations. The first circle, given by \(x^{2}+y^{2}+2 a x+c y+a=0\), transforms to reveal a center at \((-a, -\frac{c}{2})\). Similarly, the second circle \(x^{2}+y^{2}-3 a x+d y-1=0\) is centered at \((\frac{3a}{2}, -\frac{d}{2})\).

A clear understanding of these centers helps in calculating how far apart the circles are from each other, which is necessary for determining intersection conditions.

Radii of Circles

The radius of a circle refers to the distance from its center to any point on its edge. In the general circle formula \(x^2 + y^2 + 2gx + 2fy + c = 0\), the radius \(r\) is calculated using the formula \(r = \sqrt{g^2 + f^2 - c}\). This formula comes from completing the square in the circle's equation.

For the two circles provided, we find their radii respectively:

• The radius of the first circle is computed as \(\sqrt{a^2 + \left(\frac{c}{2}\right)^2 - a}\).
• For the second circle, the radius is \(\sqrt{\left(\frac{3a}{2}\right)^2 + \left(-\frac{d}{2}\right)^2 - 1}\).

The correct determination of these radii is vital to establish conditions under which the circles might intersect at two distinct points.

Distance Between Centers

To understand how two circles relate to each other spatially, it is crucial to know the distance between their centers. This distance can determine if circles intersect, touch, or remain separate. The distance \(D\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula:

\[D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]

In this particular problem, the center of the first circle is \((-a, -\frac{c}{2})\) and the second is \((\frac{3a}{2}, -\frac{d}{2})\). The distance between these centers is:

• \[D = \sqrt{\left(-a - \frac{3a}{2}\right)^2 + \left(-\frac{c}{2} + \frac{d}{2}\right)^2}\]

Knowing this distance, along with the radii of the circles, is crucial for applying the intersection conditions.

Circle Intersection Conditions

Two circles intersect at two distinct points when specific conditions relative to their centers and radii are met. For intersection to occur:

• The distance between the centers \(D\) must be less than the sum of the radii of the circles.
• The distance \(D\) must be greater than the absolute difference of their radii.

In terms of equations, if \(r_1\) and \(r_2\) are the radii of the two circles, the intersection conditions can be expressed as:

• \(D < r_1 + r_2\)
• \(D > |r_1 - r_2|\)

Applying these conditions to the circles in question ensures their intersection at exactly two distinct points. By examining these inequalities, you can determine the values of any parameters, like \(a\), necessary for intersection. This analysis reveals that the condition depends on various parameters, allowing for infinitely many solutions for \(a\) that satisfy the intersection conditions.