The number of common tangents to the circles x² + y² = 4 and x² + y² - 6x - 8y = 24 is (a) 0 (b) 1 (

Q: Wait, why do we compare the distance between centers to the difference of radii? Why not just the sum?

Great question! The sum of the radii ($R_1 + R_2$) tells us if the circles are far apart or overlapping externally. The difference ($|R_1 - R_2|$) tells us about the internal configuration. If the distance between centers is exactly the difference, it means the smaller circle is 'nestled' inside the larger one, touching it at exactly one point.

Q: I calculated the center of the second circle as (3, 4). Is that correct?

Spot on! By comparing $x^2 + y^2 - 6x - 8y - 24 = 0$ to the general form $x^2 + y^2 + 2gx + 2fy + c = 0$, we identify $2g = -6$ and $2f = -8$, leading to the center $(-g, -f) = (3, 4)$. You are moving in the right direction!

Q: If the circles touch internally, why is there only one tangent? Shouldn't there be more?

Think about the geometry: if one circle is inside the other, they can only share a tangent at the single point where their boundaries meet. Any other line would either pass through the interior of both or miss them entirely. That single point of contact acts as a 'bottleneck' for the tangent line.

Q: Is it possible for the distance between centers to be zero?

Yes, that would mean the circles are concentric (sharing the same center). In that case, they would have zero common tangents because they would never touch unless their radii were also equal, in which case they would be the same circle!

Q: Do I need to draw the graph to solve this?

While drawing is a fantastic habit for intuition, the JEE Advanced exam rewards the algebraic approach. Using the distance formula $d = |R_1 - R_2|$ is the rigorous, foolproof way to prove the configuration without relying on the accuracy of a sketch.

Question

The number of common tangents to the circles $x^2 + y^2 = 4$ and $x^2 + y^2 - 6x - 8y = 24$ is

Options:

(A) 0

(B) 1

(C) 3

(D) 4

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing on a vast coordinate plane. You have two circles, two distinct entities defined by their centers and their reach—their radii.

One circle is simple, elegant, centered at the origin $(0, 0)$ with a radius of $2$. The other is a bit more complex, shifted away from the origin, hiding its true nature behind the equation $x^2 + y^2 - 6x - 8y - 24 = 0$.

Our goal today is to determine how these two circles interact. Do they dance around each other? Do they overlap? Or do they share a singular, intimate point of contact?

### Unmasking the Circles

Before we can analyze their relationship, we must understand the individuals. The first circle, $x^2 + y^2 = 4$, is a classic. It is centered at $C_1 = (0, 0)$ with a radius $R_1 = \sqrt{4} = 2$.

Now, let us look at the second circle. The equation $x^2 + y^2 - 6x - 8y - 24 = 0$ is in the general form $x^2 + y^2 + 2gx + 2fy + c = 0$. By comparing coefficients, we find $2g = -6$ and $2f = -8$, which gives us the center $C_2 = (3, 4)$.

To find its radius, we use the formula $R_2 = \sqrt{g^2 + f^2 - c}$. Plugging in our values, we get:

$$R_2 = \sqrt{(-3)^2 + (-4)^2 - (-24)} = \sqrt{9 + 16 + 24} = \sqrt{49} = 7$$

So, we have a larger circle with a radius of $7$ centered at $(3, 4)$.

### The Distance Between Centers

Now, the tension builds. How far apart are these two centers? We use the distance formula:

$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

Substituting our coordinates, $C_1 = (0, 0)$ and $C_2 = (3, 4)$, we calculate the distance:

$$d = \sqrt{(3 - 0)^2 + (4 - 0)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$$

We have our key parameters: $R_1 = 2$, $R_2 = 7$, and the distance between centers $d = 5$.

### The Moment of Truth

This is where the physics of geometry takes over. We compare the distance $d$ with the radii. Notice something beautiful? The difference between the radii is:

$$|R_1 - R_2| = |2 - 7| = 5$$

Wait! The distance between the centers is exactly equal to the difference of the radii: $d = |R_1 - R_2|$.

In the language of geometry, when the distance between the centers of two circles is exactly equal to the difference of their radii, it means the smaller circle is sitting perfectly inside the larger one, touching it at exactly one point. They are kissing internally.

### Conclusion: The Singular Tangent

Because they touch at exactly one point, there is only one line that can be tangent to both circles at that specific location. Any other line would either cut through the circles or miss them entirely.

Thus, the number of common tangents is exactly **$1$**.

It is a profound realization, isn't it? By simply calculating the distance between two points and comparing it to the radii, we have mapped the entire relationship between these two shapes. You didn't need to draw a complex graph; you used the power of algebra to reveal the hidden truth of their interaction.

Animated Solution for Mathematics - Circles: The number of common tangents to the circles $x^{2} + y^{2} = 4$ and $x^{2} + y^{2} - 6 x - 8 y = 24$ is

Select Answer:

Visualized Solution

First Circle Equation

Center and Radius of $C_{1}$

Analyzing Circle 2

Center of $C_{2}$

Radius of $C_{2}$

Distance Between Centers

Applying Distance Formula

Calculating $C_{1} C_{2}$

Comparing Distance with Radii

Difference of Radii

Condition for Internal Touch

Number of Common Tangents

The Sigma Insight: Equation of Tangent and Normal

Analyzing the Setup

Unmasking the Circles

The Distance Between Centers

The Moment of Truth

Conclusion

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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=4 and x2+y2−6x−8y=24 is