If the curves, x² - 6x + y² + 8 = 0 and x² - 8y + y² + 16 - k = 0, (k 0) touch each other at a poi

Q: Why do we have to consider both internal and external touching? Can't we just assume they touch externally?

That is a classic trap! In coordinate geometry, when a problem states that two curves 'touch', it implies they share a common tangent at the point of contact. This can happen when one circle is outside the other (external) or when one is inside the other (internal). If you only consider one case, you miss half the potential solutions. Always check both $d = r_1 + r_2$ and $d = |r_1 - r_2|$.

Q: Why is the internal touching condition written as $|r_1 - r_2| = d$ instead of just $r_1 - r_2 = d$?

Great observation. The modulus $|r_1 - r_2|$ ensures that the distance is always positive, regardless of which radius is larger. In this specific problem, we have $r_1 = 1$ and $r_2 = \sqrt{k}$. Since we don't know if $\sqrt{k}$ is larger than $1$ or vice versa, the modulus handles both possibilities, preventing us from getting a negative distance, which is physically impossible.

Q: How do I know if $k$ can be negative? The question says $k > 0$.

The problem explicitly states $k > 0$, which simplifies things! If $k$ were allowed to be negative, the radius $\sqrt{k}$ would be imaginary, meaning the second circle wouldn't exist in the real coordinate plane. Always check the constraints given in the question; they are there to guide you.

Question

If the curves, $x^2 - 6x + y^2 + 8 = 0$ and $x^2 - 8y + y^2 + 16 - k = 0$, $(k > 0)$ touch each other at a point, then the largest value of $k$ is

Sigma Percentile · Accepted Answer

### Decoding the Circles

To understand a circle, we must find its heart—the center—and its reach—the radius. We use the method of completing the square.

For the first circle, $x^2 - 6x + y^2 + 8 = 0$, we rearrange the terms:
$$(x^2 - 6x + 9) + y^2 = -8 + 9$$

This simplifies beautifully to:
$$(x - 3)^2 + y^2 = 1$$

Here, we see the center $C_1$ is at $(3, 0)$ and the radius $r_1$ is $\sqrt{1} = 1$.

Now, let us turn our attention to the second circle: $x^2 - 8y + y^2 + 16 - k = 0$. Again, we complete the square:
$$x^2 + (y^2 - 8y + 16) = k$$

This becomes:
$$x^2 + (y - 4)^2 = k$$

The center $C_2$ is at $(0, 4)$ and the radius $r_2$ is $\sqrt{k}$. We have successfully stripped away the algebraic mask to reveal the geometric reality.

### The Distance Bridge

Now that we know where these circles live, we must find the distance between them. The distance $d$ between $C_1(3, 0)$ and $C_2(0, 4)$ is found using the distance formula:

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

This distance is the bridge connecting our two circles.

### The Geometry of Contact

The problem states that these circles touch. When two circles touch, the distance between their centers must be exactly the sum or the difference of their radii.

**Case 1: External Touching**
If they touch externally, the distance between centers is the sum of the radii: $d = r_1 + r_2$. Substituting our values, we get:
$$5 = 1 + \sqrt{k}$$

Solving for $\sqrt{k}$, we find $\sqrt{k} = 4$, which means **$k = 16$**.

**Case 2: Internal Touching**
If they touch internally, the distance between centers is the absolute difference of the radii: $d = |r_1 - r_2|$. So, $5 = |1 - \sqrt{k}|$.

This gives us two possibilities: $1 - \sqrt{k} = 5$ or $1 - \sqrt{k} = -5$. The first leads to $\sqrt{k} = -4$, which is impossible for a radius. The second leads to $\sqrt{k} = 6$, which gives **$k = 36$**.

### The Final Revelation

We have two potential values for $k$: $16$ and $36$. The question asks for the largest value of $k$.

Comparing them, we see that **$36$** is clearly the winner. By considering both the internal and external cases, we have navigated the trap that catches so many students.

Animated Solution for Mathematics - Circles: If the curves, $x^{2} - 6 x + y^{2} + 8 = 0$ and $x^{2} - 8 y + y^{2} + 16 - k = 0$ , $(k > 0)$ touch each other at a point, then the largest value of $k$ is

Enter Numerical Value:

Visualized Solution

Equation of Circle 1

Center and Radius of $C_{1}$

Equation of Circle 2

Center and Radius of $C_{2}$

Distance Between Centers

Condition for Touching Circles

Case 1: External Touching

Case 2: Internal Touching

Largest Value of $k$

The Sigma Insight: Standard and General Equation of a Circle

Decoding the Circles

The Distance Bridge

The Geometry of Contact

The Final Revelation

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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; } If the curves, x2−6x+y2+8=0 and x2−8y+y2+16−k=0, (k>0) touch each other at a point, then the largest value of k is