Problem 101 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

I. The number of common tangents to (A) 3 the circles \(x^{2}+y^{2}-6 x-2 y+9=0\) and \(x^{2}+y^{2}-14 x-8 y+61=0\) is II. The number of common tangents to (B) 4 the circles \(x^{2}+y^{2}=4\) and \(x^{2}+y^{2}-8 x\) \(+12=0\) is III. The number of common tangents to (C) 2 the circles \(x^{2}+y^{2}=4\) and \(x^{2}+y^{2}-6 x\) \(-8 y-24=0\) is IV. The number of tangents to the circle (D) 1 \(x^{2}+y^{2}-8 x-6 y+9=0\) which pass through the point \((3,-2)\) is

Step-by-Step Solution

Answer

I. 3 common tangents, II. 4 common tangents, III. 2 common tangents, IV. 1 tangent.

Step 1: Identify the Circle Equations

Recognize the standard form of the circle equation for both sets of circles. For circle (A), we have the equation \(x^2 + y^2 - 6x - 2y + 9 = 0\). For circle (B), \(x^2 + y^2 = 4\). Simplify to find the centers and radii.

Step 2: Complete the Square

For circle (A): Rewrite the equation in the form \((x-h)^2 + (y-k)^2 = r^2\). Completing the square, the equation becomes \(((x-3)^2 + (y-1)^2 = 1\). Here, the center is (3, 1) and radius is 1.

Step 3: Identify Circle Components for Circle B

Circle B is already in standard form \((x^2 + y^2 = 4)\). The circle's center is (0, 0) and radius is 2.

Step 4: Calculate the Distance between Circle Centers

For circles (A) \((x^2 + y^2 - 6x - 2y + 9 = 0)\) and (B): Calculate the distance between their centers: \( \text{Distance} = \sqrt{(3-0)^2 + (1-0)^2} = \sqrt{10} \approx 3.16\).

Step 5: Compare Distances and Radii

Calculate \(r_1+r_2 = 1+2 = 3\). If the distance between centers is greater than \(r_1 + r_2\), there are two external tangents. Since \(\sqrt{10} > 3\), these circles have 2 external tangents and no internal ones for this configuration (Total 4 tangents for disjoint circles with no overlap).

Step 6: Solve for Circles C Comparison

For circles (C) \(x^2 + y^2 = 4\) and \(x^2 + y^2 - 6x - 8y - 24 = 0\), find centers and radii as above. Calculate distance compared to sum of radii. (Details not repeated for brevity here.)

Step 7: Solving for Equation D

For equation D, first find the center and radius. The circle equation is \(x^2 + y^2 - 8x - 6y + 9 = 0\). Completing the square gives center (4,3), radius \(\sqrt{16}\). Calculate distance \((3,-2)\) from the center and analyze relative to the radius.

Step 8: Final Evaluation of Equals for D and Tangents Through Points

Compute direct tangents seeing if the external distance exceeds adding radius alone from calculated other points of tangent determined. Evaluate all cases for touches - evaluate to finalize only valid intersections being precise through geometry.

Key Concepts

Circle Equations

Understanding circle equations is foundational for solving problems involving circles. A circle's equation in standard form is \[(x-h)^2 + (y-k)^2 = r^2\]where

• \((h, k)\) is the center of the circle
• \(r\) is the radius

By seeing how these elements fit together, you can quickly identify a circle's center and radius. For instance, if we look at the equation \(x^2 + y^2 - 6x - 2y + 9 = 0\),it's crucial to rearrange it to form \((x-h)^2 + (y-k)^2 = r^2\) by completing the square. This form allows easy identification of the circle's properties, essential for plotting circles or finding tangents.

Completing the Square

Completing the square is a vital algebraic method to transform a circle's equation into its standard form. This technique helps us find the circle's center and radius with clarity.To do this, you need to organize and combine terms:

• First, take terms involving \(x\) and \(y\)
• Add and subtract necessary values to create perfect squares.

For the equation \(x^2 + y^2 - 6x - 2y + 9 = 0\), follow these steps:

• Group \(x\) and \(y\) terms: \((x^2 - 6x) + (y^2 - 2y)\)
• Complete each square by adding and subtracting \(3^2\) and \(1^2\), transforming the equation to \([ (x-3)^2 + (y-1)^2 = 1 ]\)

This process reveals that the center of the circle is \((3, 1)\) and its radius is 1.

Distance between Circle Centers

Determining the distance between two circle centers is crucial for understanding their geometric relationships, especially when calculating tangent lines. The formula to calculate this distance is derived from the distance formula:\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]Applying this formula:

• Identify centers: For circles with centers \((x_1, y_1)\) and \((x_2, y_2)\)
• Substitute values into the formula.

Considering circles \(x^2 + y^2 - 6x - 2y + 9 = 0\) and \(x^2 + y^2 = 4\),their centers are \((3, 1)\) and \((0, 0)\),respectively. Using the distance formula, we find:\[ \text{Distance} = \sqrt{(3-0)^2 + (1-0)^2} = \sqrt{10} \approx 3.16 \]Knowing this distance helps to ascertain whether circles are overlapping, disjoint, or tangent to each other.

External and Internal Tangents

Tangents are lines that touch circles at exactly one point. Given two circles, there can be external and internal tangents.**External Tangents**:

• These tangents lie outside the circles
• Only touch circles at separate points

To find external tangents:

• Calculate the sum of the radii ( \(r_1 + r_2\))
• Check if the distance between centers is greater than this sum

For example, circles with centers \((3, 1)\) and \((0, 0)\) with radii \(1\) and \(2\)will have external tangents if \(\sqrt{10} > 3\).**Internal Tangents**:Occur when the distance between centers is less, necessitating different geometry.Understanding tangent properties is crucial in many geometric problems, helping visualize and calculate precise circle interactions with lines.