Problem 73 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If \(P\) and \(Q\) are the points of intersection of the circles \(x^{2}+y^{2}+3 x+7 y+2 p-5=0\) and \(x^{2}+y^{2}+2 x+2 y-p^{2}=0\) then there is a circle passing through \(\mathrm{P}, \mathrm{Q}\) and \((1,1)\) for: (a) all except one value of \(p\) (b) all except two values of \(p\) (c) exactly one value of \(p\) (d) all values of \(p\)

Step-by-Step Solution

Answer

The answer is (d): all values of \( p \). The additional circle through $(1,1)$ satisfies every instance.

Step 1: Identify Intersection Points

We need to find the intersection points of the given circles. This involves subtracting the equations to eliminate terms and simplify the resulting equation. First, we have:\[ x^2 + y^2 + 3x + 7y + 2p - 5 = 0 \] and \[ x^2 + y^2 + 2x + 2y - p^2 = 0. \] Subtracting these, we obtain: \[ (3x + 7y + 2p - 5) - (2x + 2y - p^2) = 0 \] which simplifies to: \[ x + 5y + 2p - p^2 - 5 = 0. \] This equation must be satisfied by the points of intersection \(P\) and \(Q\).

Step 2: Equation of Circle Through Points

We want to find a circle that passes through both points \(P\), \(Q\) (intersection points) and the point \((1,1)\). The equation of a circle in the general form passing through any points can be written as (considering it must satisfy known points and conditions): \[ (x - 1)^2 + (y - 1)^2 = r^2 \] where \((1,1)\) is one of the points. We aim to relate this to the found equation for intersections.

Step 3: Codifying the Condition for Values of p

We need to relate \(p\) such that the derived equation of the circle through \((1,1)\) remains valid without contradicting the existence of intersection points of the original set of circles. The line equation \( x + 5y + 2p - p^2 - 5 = 0 \) suggests a relationship. We want to find for how many \(p\) this relationship might fail to enlist a third unique point as part of an intersecting family of circles coaxially modifying through \((1,1)\). This boils down to solving by constraint values, considering the geometric intersection implications for all values of \(p\).

Step 4: Analyze Different Cases and Outcomes

By assessing the line equation solutions directly through intersection analysis of circles, it's required the system will (a) typically not satisfy for a particular critical value where an edge tangent transition (the common line) may arise. Thus such problem revolves around confirming cases leading to either singular or none overlapping addition to circles in prescribed (multi-family) geometry.

Key Concepts

circles equation

A circle's equation can be expressed in two principal forms: the standard form \( (x-h)^2 + (y-k)^2 = r^2 \), where \((h,k)\) is the center and \(r\) is the radius; and the general form \( x^2 + y^2 + ax + by + c = 0 \).

In the original problem, we use the general form, where two equations describe circles whose points of intersection we are interested in. To find these points, we work with their equations:

• \( x^2 + y^2 + 3x + 7y + 2p - 5 = 0 \)
• \( x^2 + y^2 + 2x + 2y - p^2 = 0 \)

These equations involve constants and variable \(p\) that can affect the position and intersection characteristics of the circles. Understanding these equations is essential in finding the regions and points where the circles meet.

intersection points

Intersection points of two circles are the locations where both circles share common points.

When given two circles, finding these points involves solving their equations simultaneously. In our case, this process begins by subtracting one circle's equation from the other.

This subtraction helps cancel the identical terms: \( x^2 \) and \( y^2 \). This leads to:\( x + 5y + 2p - p^2 - 5 = 0 \).This line equation represents a constraint the intersection points must satisfy, indicating where these two circles meet. However, important detail is that, sometimes, intersection might not exist or could mean overlapping circles.

solving simultaneous equations

Simultaneous equations are, essentially, sets of equations that are solved together.

The roots for these can represent coordinates of intersection points of circles or lines.

In the original exercise, the two circle equations:

• \( x^2 + y^2 + 3x + 7y + 2p - 5 = 0 \)
• \( x^2 + y^2 + 2x + 2y - p^2 = 0 \)

are simplified by subtraction to isolate the key constraints:\( x + 5y + 2p - p^2 - 5 = 0 \).This means that solving simultaneous equations here requires examining feasible \(p\) values such that a valid x, y pair is obtained, indicating the interaction between these circles.

geometry of circles

The geometry of circles covers aspects like position, distance, and intersection.

The main focus in the problem is the points of intersection. Think of these points as physical places where the boundary of two circles visually cross. These are crucial in determining third-party circle locations.

• Understanding the center and radius allows us to imagine intersecting and transitioning circles.
• In this context, consider circles forming specific geometric arrangements, like being tangent or concentric.

The problem implicitly asks for configuring intersecting circles to pass through new predetermined points \((1,1)\). Thus, the exploration aims to discover conditions on \(p\) that permit such mathematical constructs in the coordinate plane.