The locus of the centre of a circle, which touches externally the circle x² + y² - 6x - 8y + 14 = 0

Q: Wait, why is the radius $r = h$?

Great question! When a circle touches the y-axis, the perpendicular distance from its center $(h, k)$ to the y-axis is simply the absolute value of its x-coordinate, $|h|$. Since the circle is touching the axis, this distance is exactly the radius $r$. We assume $h > 0$ for the circle to be on the right side, so $r = h$.

Q: Why does external touch mean the distance between centers is $r_1 + r_2$?

Think of it geometrically. If two circles are outside each other and just touching at a single point, the line segment connecting their centers passes through that point of contact. The length of this segment is the sum of the radius of the first circle and the radius of the second circle. It is a fundamental property of tangent circles!

Q: Why do we replace $(h, k)$ with $(x, y)$ at the end?

The coordinates $(h, k)$ were just a temporary label for the center of our moving circle. To define the path (the locus) that this center traces, we need an equation in terms of the general coordinate variables $x$ and $y$. Replacing $(h, k)$ with $(x, y)$ gives us the final equation of the curve that the center follows.

Question

The locus of the centre of a circle, which touches externally the circle $x^2 + y^2 - 6x - 8y + 14 = 0$ and also touches the y-axis, is given by the equation:

Options:

(A) $x^2 - 6x - 10y + 14 = 0$

(B) $x^2 - 10x - 6y + 14 = 0$

(C) $y^2 - 6x - 10y + 14 = 0$

(D) $y^2 - 10x - 6y + 14 = 0$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing on the coordinate plane, watching a circle dance. This circle is not just sitting still; it is moving, and its center, which we will call $(h, k)$, is tracing a path known as the locus.

Our circle has two strict rules to follow: it must always touch the y-axis, and it must always touch a fixed circle externally.

If a circle touches the y-axis, the perpendicular distance from its center $(h, k)$ to the y-axis must be equal to its radius $r$. Since the x-coordinate of the center is $h$, the radius is simply $r = |h|$.

Assuming the circle lies on the right side of the y-axis, we have the primary constraint:
$$r = h$$

### Decoding the Fixed Anchor

Now, let us analyze the fixed circle given by the equation:
$$x^2 + y^2 - 6x - 6y + 14 = 0$$

By comparing this to the general form $x^2 + y^2 + 2gx + 2fy + c = 0$, we identify the center $C_1$ and the radius $r_1$:
$$C_1 = (-g, -f) = (3, 3)$$
$$r_1 = \sqrt{g^2 + f^2 - c} = \sqrt{3^2 + 3^2 - 14} = \sqrt{18 - 14} = \sqrt{4} = 2$$

We now have a fixed anchor at $(3, 3)$ with a radius of $2$.

### The Dance of the Centers

When two circles touch each other externally, the distance between their centers is exactly equal to the sum of their radii.

The distance between our moving center $(h, k)$ and the fixed center $(3, 3)$ must be $r + r_1$. Substituting our known values, we obtain the distance equation:
$$\sqrt{(h-3)^2 + (k-3)^2} = h + 2$$

### The Algebraic Transformation

To eliminate the square root, we square both sides of the equation:
$$(h-3)^2 + (k-3)^2 = (h+2)^2$$

Expanding both sides yields:
$$(h^2 - 6h + 9) + (k^2 - 6k + 9) = h^2 + 4h + 4$$

Notice that the $h^2$ terms on both sides cancel out perfectly. Simplifying the remaining terms, we get:
$$-6h + 18 + k^2 - 6k = 4h + 4$$
$$k^2 - 6k - 10h + 14 = 0$$

### The Locus Revealed

To find the final equation of the locus, we replace the temporary coordinates $(h, k)$ with the general variables $(x, y)$.

The resulting equation of the path is:
$$y^2 - 6y - 10x + 14 = 0$$

This is the equation of a **parabola**. You have successfully tracked the center of the moving circle and discovered its path.

Animated Solution for Mathematics - Circles: The locus of the centre of a circle, which touches externally the circle $x^{2} + y^{2} - 6 x - 8 y + 14 = 0$ and also touches the y-axis, is given by the equation:

Select Answer:

Visualized Solution

Visualizing the Problem

Defining the Moving Circle

Analyzing the Fixed Circle

Condition for External Touch

Squaring Both Sides

Expanding the Terms

Simplifying the Equation

Rearranging to Standard Form

Final Locus Equation

The Sigma Insight: Standard and General Equation of a Circle

Analyzing the Setup

Decoding the Fixed Anchor

The Dance of the Centers

The Algebraic Transformation

The Locus Revealed

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