Problem 17 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The common tangent to the circles \(x^{2}+y^{2}=4\) and \(x^{2}+y^{2}+\) \(6 x+8 y-24=0\) also passes through the point: (a) \((4,-2)\) (b) \((-6,4)\) (c) \((6,-2)\) (d) \((-4,6)\)

Step-by-Step Solution

Answer

None of the given points lies on the common tangent line as computed.

Step 1: Identify the circle equations and centers

The first circle equation is given as \(x^{2}+y^{2}=4\), which has its center at \((0, 0)\) and radius \(2\). The second circle equation can be rewritten as \(x^2 + y^2 + 6x + 8y = 24\). Complete the square for both \(x\) and \(y\) terms to find the center: \((x+3)^2 + (y+4)^2 = 49\), resulting in center \((-3, -4)\) and radius \(7\).

Step 2: Find the equation of the common tangent

The general equation of the tangent to a circle \((x-h)^2 + (y-k)^2 = r^2\) is \( (x - h)\cos\theta + (y - k)\sin\theta = r\). For the first circle, substitute center \((0, 0)\) and radius \(2\) to get: \(x\cos\theta + y\sin\theta = 2\). For the second circle, substitute center \((-3, -4)\) and radius \(7\) to get: \((x+3)\cos\theta + (y+4)\sin\theta = 7\), which simplifies to \(x\cos\theta + y\sin\theta = 7 - 3\cos\theta - 4\sin\theta\).

Step 3: Equate and solve for common tangent condition

Set the tangent equations from each circle equal to each other: \(2 = 7 - 3\cos\theta - 4\sin\theta\). Solving gives \(3\cos\theta + 4\sin\theta = 5\). This is the line equation of the common tangent for both circles.

Step 4: Verify which point lies on the common tangent line

Substitute each given point into \(3x + 4y = 5\) to determine if it satisfies the equation. Check point by point:- For \((4, -2)\), substitution yields \(3(4) + 4(-2) = 12 - 8 = 4\), not equal to 5.- For \((-6, 4)\), substitution gives \(3(-6) + 4(4) = -18 + 16 = -2\), not equal to 5.- For \((6, -2)\), substitution shows \(3(6) + 4(-2) = 18 - 8 = 10\), not equal to 5.- For \((-4, 6)\), substitution yields \(3(-4) + 4(6) = -12 + 24 = 12\), not equal to 5.

Step 5: Identify correct answer

After computation, it appears none of the given options satisfy the tangent line equation \(3x + 4y = 5\). Hence, the provided points in the options do not lie on the common tangent as calculated. The exercise seems to be based on incorrect options or there may have been an error in calculation.

Key Concepts

Circle Equations

Circle equations are fundamental in coordinate geometry as they define the geometric shape perfectly in the Cartesian plane. A standard form of a circle's equation is

• \((x - h)^2 + (y - k)^2 = r^2\), where:
  • \((h, k)\) represents the center of the circle,
  • \(r\) is the radius.

For example, the equation \(x^2 + y^2 = 4\) represents a circle centered at \((0,0)\) with a radius of 2. On the other hand, the equation \(x^2 + y^2 + 6x + 8y - 24 = 0\) requires completing the square to identify the center and radius accurately, transforming it to

• \((x + 3)^2 + (y + 4)^2 = 49\), indicating a center at \((-3, -4)\) and a radius of 7.

Understanding and manipulating these equations allows you to plot circles accurately in a coordinate system.

Common Tangent

A common tangent to circles is a line that touches two circles at exactly one point each. Finding the common tangent involves:

• Writing the tangent equation for each circle.
• Equating them to establish the relationship or common line that satisfies being tangent to both circles.

For instance, the tangent equation of a circle

• \((x-h)^2+(y-k)^2 = r^2\)

can be expressed as:

• \((x - h) ext{cos} heta + (y - k) ext{sin} heta = r\).

By solving these equations for the two given circles, we can find the equation \(3\text{cos} heta + 4\text{sin} heta = 5\), which serves as a common tangent.

Coordinate Geometry

Coordinate geometry, or analytic geometry, combines algebra and geometry through a system of coordinates. The primary usefulness lies in its ability to describe geometric figures using algebraic equations, providing:

• A method to solve geometrical problems with algebraic systems.
• The representation of points, lines, and curves in the plane using a coordinate system.

In the context of circles, coordinate geometry is crucial for:

• Deriving the positions of their centers,
• Determining points of tangency,
• And exploring intersections with other geometric figures as lines or other circles.

This interconnectedness allows for the simple understanding and manipulation of geometric entities using x-y coordinate axes.

Analytical Geometry

Analytical geometry is an extension of coordinate geometry where analytics and algebra work together to solve geometry problems. It plays an essential role in understanding:

• The relationships between different geometric figures using equations.
• The analytical derivation of distances, slopes, tangents, and more.

In exercises involving circles and tangents, analytical geometry helps to

• Establish precise measurements and relationships between the circles’ centers and tangents' slopes.
• For example, it guides the step-by-step process of identifying tangent equations, as shown by determining the equation \(3x + 4y = 5\) as a common tangent.

This effective approach allows for a deeper understanding of geometry using computation and logic.