If the circle x² + y² - 2gx + 6y - 19c = 0, g, c ∈ ℝ passes through the point (6, 1) and its centre

Q: Wait, why is the center $(g, -3)$ and not $(-g, 3)$? I always get the signs wrong!

This is the most common trap in coordinate geometry! Always remember the standard form: $x^2 + y^2 + 2gx + 2fy + c = 0$. The center is $(-g, -f)$. In your equation, the term is $-2gx$. Comparing this to $+2gx_{std}$, we see that $2g_{std} = -2g$, so $g_{std} = -g$. Thus, the x-coordinate of the center is $-(-g) = g$. Similarly, for the $y$ term, we have $+6y$, so $2f_{std} = 6$, meaning $f_{std} = 3$. The y-coordinate of the center is $-f_{std} = -3$. Always map the coefficients carefully!

Q: Why can't we just find the intercept directly without solving for $g$ and $c$?

That is a brilliant intuition, but look at the equation: $x^2 + y^2 - 2gx + 6y - 19c = 0$. The intercept depends on both $g$ and $c$. Without knowing these specific values, the circle is not 'fixed' in the plane. It is a family of circles. The problem provides two constraints specifically to 'lock' the circle into one unique position. You must solve for the parameters to define the circle before you can measure its properties.

Q: Is the intercept formula always $2\sqrt{g^2 - c}$? I am confused by the variables.

Great catch! The formula $2\sqrt{g^2 - c}$ uses the $g$ and $c$ from the *standard* equation $x^2 + y^2 + 2gx + 2fy + c = 0$. In your specific problem, the equation has different coefficients. It is safer to think of the formula as $2\sqrt{(\text{half of x-coefficient})^2 - (\text{constant term})}$. Always derive the values from your specific equation rather than memorizing the variables blindly.

Question

If the circle $x^2 + y^2 - 2gx + 6y - 19c = 0, g, c \in \mathbb{R}$ passes through the point $(6, 1)$ and its centre lies on the line $x - 2cy = 8$, then the length of intercept made by the circle on x-axis is

Options:

(A) $\sqrt{11}$

(B) 4

(C) 3

(D) $2\sqrt{23}$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

My dear student, welcome to the world of coordinate geometry. Today, we are not just solving an equation; we are performing a detective operation.

We have a circle, a mysterious entity defined by the equation $x^2 + y^2 - 2gx + 6y - 19c = 0$. It is currently floating in the Cartesian plane, its position and size dictated by two unknown parameters, $g$ and $c$.

Our mission is to pin this circle down, find its true identity, and calculate the length of the intercept it carves out on the x-axis.

### Phase 1: The Power of Substitution

Imagine you are standing on the coordinate plane. You are told that the point $(6, 1)$ lies exactly on the circumference of this circle.

In the language of mathematics, this is a command. It means that if you plug $x = 6$ and $y = 1$ into the equation, the equality must hold true.

Let us perform this substitution with precision:

$$6^2 + 1^2 - 2g(6) + 6(1) - 19c = 0$$

Now, let us simplify this. $6^2$ is $36$, and $1^2$ is $1$. Adding the $6$ from the $y$ term, we get $36 + 1 + 6 = 43$.

The equation transforms into:

$$43 - 12g - 19c = 0$$

Rearranging this, we arrive at our first vital clue, which we shall call **Equation 1**:

$$12g + 19c = 43$$

### Phase 2: The Heart of the Circle

Every circle has a heart—its center. To find it, we look at the general form $x^2 + y^2 + 2gx_{std} + 2fy_{std} + c_{const} = 0$.

By comparing our equation $x^2 + y^2 - 2gx + 6y - 19c = 0$ to this standard form, we identify the center. The x-coordinate is half the coefficient of $x$ with the sign flipped, which gives us $g$.

The y-coordinate is half the coefficient of $y$ with the sign flipped, which gives us $-3$. Thus, our center is at $(g, -3)$.

But the problem gives us another constraint: this center lies on the line $x - 2cy = 8$. If the center $(g, -3)$ lies on this line, it must satisfy the line's equation.

Let us substitute:

$$g - 2c(-3) = 8$$

Simplifying this, we get:

$$g + 6c = 8$$

This is **Equation 2**. We now have a system of two linear equations. The fog is lifting.

### Phase 3: The Algebraic Resolution

We have our system:

1) $12g + 19c = 43$

2) $g = 8 - 6c$

Let us substitute the expression for $g$ from the second equation into the first.

$$12(8 - 6c) + 19c = 43$$

Expanding the brackets, we get $96 - 72c + 19c = 43$. Combining the $c$ terms, $-72c + 19c$ gives us $-53c$.

So:

$$96 - 53c = 43$$

Subtracting $96$ from both sides, we get $-53c = -53$. Dividing by $-53$, we find the elegant result: $c = 1$.

With $c = 1$ in our pocket, finding $g$ is trivial. Using $g = 8 - 6c$, we get $g = 8 - 6(1) = 2$.

We have successfully identified our circle: $x^2 + y^2 - 4x + 6y - 19 = 0$.

### Phase 4: The Grand Finale

We have arrived at the final stage. We need the length of the x-intercept. The formula for the x-intercept of a circle $x^2 + y^2 + 2gx_{std} + 2fy_{std} + c_{const} = 0$ is $2\sqrt{g_{std}^2 - c_{const}}$.

Looking at our circle $x^2 + y^2 - 4x + 6y - 19 = 0$, we identify $g_{std} = -2$ and $c_{const} = -19$. Plugging these into our formula:

$$	ext{Intercept} = 2\sqrt{(-2)^2 - (-19)}$$

$$	ext{Intercept} = 2\sqrt{4 + 19}$$

$$	ext{Intercept} = 2\sqrt{23}$$

And there it is! The length of the intercept is **$2\sqrt{23}$**.

You see, geometry is not just about shapes; it is about the logical unfolding of constraints. You started with unknowns, you applied the laws of algebra, and you arrived at a concrete, beautiful truth.

Animated Solution for Mathematics - Circles: If the circle $x^{2} + y^{2} - 2 g x + 6 y - 19 c = 0, g, c \in R$ passes through the point $(6, 1)$ and its centre lies on the line $x - 2 cy = 8$ , then the length of intercept made by the circle on x-axis is

Select Answer:

Visualized Solution

Problem Setup

Substituting Point $(6, 1)$

Simplifying Equation 1

Identifying the Center

Center on the Given Line

Simplifying Equation 2

Solving for $c$

Solving for $g$

The X-Intercept Formula

Final Calculation

The Sigma Insight: Intercepts Made by a Circle

Analyzing the Setup

Phase 1

Phase 2

Phase 3

Phase 4

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