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Animated Solution for Mathematics - Circles: The number of common tangents to the circles $x^{2} + y^{2} - 4 x - 6 y - 12 = 0$ and $x^{2} + y^{2} + 6 x + 18 y + 26 = 0$ , is :

Q: How do I know if they touch externally or internally?

Great question! If the distance between centers $d = r_1 + r_2$, they touch externally (like two coins side-by-side). If $d = |r_1 - r_2|$, they touch internally (one inside the other). In our case, $d = 13$ and $r_1 + r_2 = 13$, so they are definitely external.

Select Answer:

Visualized Solution

The Geometry of Common Tangents

Given two circles, we need to find the number of common tangents.
The number of tangents depends on the relative position of the circles.

Analyzing Circle 1

First circle: $x^{2} + y^{2} - 4 x - 6 y - 12 = 0$
Compare with general form: $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$
Center $C_{1} = (- g, - f)$
Radius $r_{1} = g^{2} + f^{2} - c$

Center and Radius of Circle 1

$2 g = - 4 ⟹ g = - 2$
$2 f = - 6 ⟹ f = - 3$
Center $C_{1} = (2, 3)$
$r_{1} = (- 2)^{2} + (- 3)^{2} - (- 12) = 4 + 9 + 12 = 5$

Analyzing Circle 2

Second circle: $x^{2} + y^{2} + 6 x + 18 y + 26 = 0$
Center $C_{2} = (- g, - f)$
Radius $r_{2} = g^{2} + f^{2} - c$

Center and Radius of Circle 2

$2 g = 6 ⟹ g = 3$
$2 f = 18 ⟹ f = 9$
Center $C_{2} = (- 3, - 9)$
$r_{2} = 3^{2} + 9^{2} - 26 = 9 + 81 - 26 = 64 = 8$

Distance Between Centers

To find relative position, calculate distance $d$ between $C_{1} (2, 3)$ and $C_{2} (- 3, - 9)$ .
Distance formula: $d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$

Substituting Coordinates

$d = (- 3 - 2)^{2} + (- 9 - 3)^{2}$

Calculating the Distance

$d = (- 5)^{2} + (- 12)^{2}$
$d = 25 + 144$
$d = 169 = 13$

Sum of Radii

We have $d = 13$ .
Now, let's find the sum of their radii: $r_{1} + r_{2}$ .
$r_{1} + r_{2} = 5 + 8 = 13$

Condition for External Touching

Observation: $d = r_{1} + r_{2}$ ( $13 = 13$ )
Conclusion: The circles touch each other externally at exactly one point.

Transverse Common Tangent

Because they touch at one point, we can draw a tangent passing right between them.
This is called the Transverse Common Tangent.
Number of transverse tangents = $1$ .

Direct Common Tangents

We can also draw tangents that touch both circles on the outside without crossing the line segment joining their centers.
These are called Direct Common Tangents.
Number of direct tangents = $2$ .

Total Number of Common Tangents

Transverse Tangents = $1$
Direct Tangents = $2$
Total Common Tangents = $1 + 2 = 3$
The correct option is 3.

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The Sigma Insight: Standard and General Equation of a Circle

Solution Diagram

Analyzing the Setup

To determine the number of common tangents between two circles, we must first identify their centers and radii. We begin by analyzing the first circle:

x^{2} + y^{2} - 4 x - 6 y - 12 = 0

Comparing this to the general form

x^{2} + y^{2} + 2 g x + 2 f y + c = 0

, we identify

g = - 2

and

f = - 3

. Thus, the center is

C_{1} = (2, 3)

The radius

r_{1}

is calculated as follows:

r_{1} = g^{2} + f^{2} - c = (- 2)^{2} + (- 3)^{2} - (- 12) = 4 + 9 + 12 = 25 = 5

Unmasking the Second Circle

Next, we examine the second circle:

x^{2} + y^{2} + 6 x + 18 y + 26 = 0

Following the same logic, we identify

g = 3

and

f = 9

, which gives us the center

C_{2} = (- 3, - 9)

. The radius

r_{2}

is:

r_{2} = 3^{2} + 9^{2} - 26 = 9 + 81 - 26 = 64 = 8

We now have our primary parameters:

C_{1} (2, 3)

with

r_{1} = 5

, and

C_{2} (- 3, - 9)

with

r_{2} = 8

The Bridge Between Centers

To understand the relative position of these circles, we calculate the distance

d

between their centers

C_{1}

and

C_{2}

using the distance formula:

d = (- 3 - 2)^{2} + (- 9 - 3)^{2} = (- 5)^{2} + (- 12)^{2} = 25 + 144 = 169 = 13

We compare this distance to the sum of the radii:

r_{1} + r_{2} = 5 + 8 = 13

Since

d = r_{1} + r_{2}

, the distance between the centers is exactly equal to the sum of the radii. This indicates that the two circles are touching externally at a single point.

Final Calculation

When two circles touch externally, the number of common tangents is determined by their configuration:

1. There is one transverse common tangent that passes through the point of contact. 2. There are two direct common tangents that wrap around the exterior of both circles.

Summing these, we find the total number of common tangents is

1 + 2 = 3

The two circles share exactly 3 common tangents.

Common Traps & FAQs

Wait, why do we calculate the center and radius first? Can't we just use the tangent condition directly?

You could, but that would be like trying to solve a maze by running into every wall! By finding the center and radius, you are essentially mapping out the 'territory' of the circles. It simplifies the problem from an algebraic nightmare into a simple geometric comparison. Always look for the geometric shortcut first!

How do I know if they touch externally or internally?

Great question! If the distance between centers

d = r_{1} + r_{2}

, they touch externally (like two coins side-by-side). If

d = ∣ r_{1} - r_{2} ∣

, they touch internally (one inside the other). In our case,

d = 13

and

r_{1} + r_{2} = 13

, so they are definitely external.

Why is the answer 3 and not 4? I thought two circles always have 4 tangents.

That is a common trap! Two circles have 4 common tangents only when they are completely separate (

d > r_{1} + r_{2}

). When they touch externally, the two transverse tangents merge into one, reducing the total count to 3. It is all about how many lines can 'hug' both circles simultaneously.

Animated Solution for Mathematics - Circles: The number of common tangents to the circles $x^{2} + y^{2} - 4 x - 6 y - 12 = 0$ and $x^{2} + y^{2} + 6 x + 18 y + 26 = 0$ , is :

Select Answer:

Visualized Solution

The Geometry of Common Tangents

Analyzing Circle 1

Center and Radius of Circle 1

Analyzing Circle 2

Center and Radius of Circle 2

Distance Between Centers

Substituting Coordinates

Calculating the Distance

Sum of Radii

Condition for External Touching

Transverse Common Tangent

Direct Common Tangents

Total Number of Common Tangents

The Sigma Insight: Standard and General Equation of a Circle

Analyzing the Setup

Unmasking the Second Circle

The Bridge Between Centers

Final Calculation

Similar Questions

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Animated Solution for Mathematics - Circles: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of common tangents to the circles x2+y2−4x−6y−12=0 and x2+y2+6x+18y+26=0, is :

Select Answer:

Visualized Solution

The Geometry of Common Tangents

Analyzing Circle 1

Center and Radius of Circle 1

Analyzing Circle 2

Center and Radius of Circle 2

Distance Between Centers

Substituting Coordinates

Calculating the Distance

Sum of Radii

Condition for External Touching

Transverse Common Tangent

Direct Common Tangents

Total Number of Common Tangents

The Sigma Insight: Standard and General Equation of a Circle

Analyzing the Setup

Unmasking the Second Circle

The Bridge Between Centers

Final Calculation

Similar Questions

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Choose the correct statement about two circles whose equations are given below: x2+y2−10x−10y+41=0 and x2+y2−22x−10y+137=0

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Animated Solution for Mathematics - Circles: The number of common tangents to the circles $x^{2} + y^{2} - 4 x - 6 y - 12 = 0$ and $x^{2} + y^{2} + 6 x + 18 y + 26 = 0$ , is :

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If one of the diameters o of the circle $x^{2} + y^{2} - 2 x - 6 y + 6 = 0$ is a chord of another circle 'C' whose center is at (2, 1), then its radius is

If one of the diameters of the circle, given by the equation, $x^{2} + y^{2} - 4 x + 6 y - 12 = 0$ , is a chord of a circle $S$ , whose centre is at $(- 3, 2)$ , then the radius of $S$ is: