Problem 82 - A Complete Resource Book in Mathematics for JEE Main

Question

For the circles \(S_{1}=x^{2}+y^{2}-4 x-6 y-12=0\) and \(S_{2}=\) \(x^{2}+y^{2}+6 x+4 y-12=0\) and the line \(L=x+y=0\) (A) \(L\) is the common tangent of \(S_{1}\) and \(S_{2}\) (B) \(L\) is the common chord of \(S_{1}\) and \(S_{2}\) (C) \(L\) is radical axis of \(S_{1}\) and \(S_{2}\) (D) \(L\) is perpendicular to the line joining the centres of \(S_{1}\) and \(S_{2}\)

Step-by-Step Solution

Answer

The line \(L\) is the radical axis and is perpendicular to the line joining the centers.

Step 1: Complete the Square for Circle Equations

For the circle equation \( S_1: x^2 + y^2 - 4x - 6y - 12 = 0 \), we complete the square:\[ (x - 2)^2 + (y - 3)^2 = 25 \]So, the center of \( S_1 \) is \((2, 3)\) and radius is \(5\).For the equation \( S_2: x^2 + y^2 + 6x + 4y - 12 = 0 \),\[ (x + 3)^2 + (y + 2)^2 = 25 \]So, the center of \( S_2 \) is \((-3, -2)\) and radius is \(5\).

Step 2: Determine Line Relationship

The line \( L: x + y = 0 \) can be rewritten in point-slope form as \( y = -x \).Substitute any point on \( L \) such as \((2, -2)\) or \( (-3, 3)\). The line passes through neither of the circle centers and does not affect the radius length directly.

Step 3: Identify the Role of Line L

To check if \( L \) is a common tangent, compute the distance from each center to the line \( L \), which should equal the radius (5).Distance formula for a point \((x_1, y_1)\) to \( ax + by + c = 0 \) is\[ \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}} \]Computing for \( (2, 3) \) and \( (-3, -2) \), neither distance equals 5, hence \( L \) is not a common tangent.

Step 4: Calculate and Compare Powers of the Line

Compute the power of the line with respect to each circle:For \( S_1 \), power = \((2)^2 + (3)^2 - 12 = 13 - 12 = 1 \).For \( S_2 \), power = \((-3)^2 + (-2)^2 - 12 = 13 - 12 = 1 \).Equal powers imply that \( L \) is the radical axis of \( S_1 \) and \( S_2 \), confirming option (C).

Step 5: Check Perpendicularity to Center Line

The slope of the line joining the centers \((2, 3)\) and \((-3, -2)\) is \( m = 1 \), while the slope of \( L \) is \(-1\), indicating perpendicularity (as slopes are negative reciprocals). This confirms option (D).

Key Concepts

Tangent Lines

A tangent line to a circle is a straight line that just "touches" the circle at exactly one point. Imagine a point on the edge of the circle where the line only grazes the circle without actually cutting through it. For a line to be a tangent, the distance from the center of the circle to the line must equal the circle's radius. This is a crucial idea when determining whether a line is a tangent to a given circle. To check for tangency between a line and a circle:

• Find the distance from the center of the circle to the line using the distance formula for a point to a line, which is given by \( \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}} \) where \( x_1 \) and \( y_1 \) are the coordinates of the center of the circle.
• Compare this distance with the circle's radius.

If they are equal, the line is tangent to the circle. In our exercise, line \( L \) was tested against this criterion for both circles \( S_1 \) and \( S_2 \), and neither distance matched the radius, so \( L \) was not a common tangent.

Circle Equations

Circle equations typically come in the standard form \( (x - h)^2 + (y - k)^2 = r^2 \), where \((h, k)\) is the center of the circle, and \(r\) is the radius. This equation arises from the distance formula, expressing that every point \((x, y)\) on the circle is exactly \(r\) units away from the center \((h, k)\). Completing the square is often essential to converting general circle equations into standard form:

• Identify and group the \( x \) and \( y \) terms.
• Complete the square for both sets of terms to find the center \((h, k)\).
• Adjust the equation to reveal the radius squared \( r^2 \).

In the given exercise, the original equations for \( S_1 \) and \( S_2 \) were modified using these steps, identifying their respective centers and radii.

Line Relationships

Understanding how lines interact with circles requires examining several possible relationships, such as tangency, intersections, or distinct separation. The line \( L: x+y=0 \), for instance, is described in slope-intercept form as \( y=-x \). This describes a line through the origin with a slope of \(-1\), a direct line across the first and third quadrants of the coordinate plane.To explore line and circle relationships, consider:

• Whether the line intersects, is tangent, or disjoint from the circle;
• Projection of either circle center onto the line, using the perpendicular distance metric;
• Slope comparisons between different segments related to the circle and the line.

In this exercise, the line \( L \) was neither intersecting both circles nor serving as a tangent, producing unique interactions further explored in the context of the radical axis.

Radical Axis

The radical axis of two circles is an important geometric concept representing the locus of points having equal power with respect to the two circles. Power of a point in circle geometry is defined by the difference between the squared distances from the point to the circle center and the square of the circle's radius.To find the radical axis:

• Compute the power of the line with respect to each circle.
• If these powers are equal, the line is the radical axis.

In our problem, line \( L \) demonstrated equal power for both circles \( S_1 \) and \( S_2 \), confirming its status as their radical axis. This indicates that any point along \( L \) maintains a unique equidistance-like property, marking it as a line of significance in circle geometry.

Circle Centers

Circle centers are fundamental in defining a circle's geometry. They serve as the "anchor point" from which the circle's perimeter extends equally in all directions. In Cartesian equations, the center coordinates \((h, k)\) appear once the equation is in its standard form. Calculating a circle's center often involves completing the square to restructure the general circle equation. Important aspects regarding circle centers include:

• Determining the coordinates \((h, k)\) by examining completed square terms;
• Using these centroids to assess line relationships like tangency or perpendicularity;
• Measuring interactions and distances between multiple center points.

For the circles \( S_1 \) and \( S_2 \), centers \( (2, 3) \) and \((-3, -2)\) respectively not only set the circles' locations but also allowed examinations of how \( L \) interacts with the line joining these centers, highlighting perpendicularity in our tasks.