Circle Intercept On X-axis: Find The Length

Hey guys, let's dive into a cool geometry problem today, focusing on circles and their intercepts on the X-axis. We've got this circle equation: x² + y² - 2gx + 6y - 19c = 0, where 'c' is a real number. This circle isn't just any circle; it passes through a specific point, (6, 1), and its center chills on the line x - 2cy = 8. Our mission, should we choose to accept it, is to find the length of the intercept this circle makes on the X-axis. Get ready to flex those math muscles because this is gonna be a fun ride!

Understanding the Circle's Properties

Alright, first things first, let's get a handle on the standard form of a circle's equation. The general form we're given is x² + y² - 2gx + 6y - 19c = 0. To find the center and radius, we usually want to convert this to the standard form: (x - h)² + (y - k)² = r², where (h, k) is the center and 'r' is the radius. By comparing the general form to the standard form, we can deduce some key information. The center of our circle, in terms of 'g' and the coefficients of y, is at (g, -3). Now, this is where things get interesting. We're told the center (g, -3) lies on the line x - 2cy = 8. Plugging in the coordinates of the center into the line's equation, we get g - 2c(-3) = 8, which simplifies to g + 6c = 8. This equation gives us a relationship between 'g' and 'c', which will be super useful later on. Remember, we're looking for the length of the intercept on the X-axis, and that depends heavily on the circle's center and radius.

Using the Point of Passage

Now, the problem states that the circle passes through the point (6, 1). This is a crucial piece of information that allows us to substitute these coordinates into the circle's general equation. So, let's plug in x = 6 and y = 1 into x² + y² - 2gx + 6y - 19c = 0:

(6)² + (1)² - 2g(6) + 6(1) - 19c = 0

This simplifies to:

36 + 1 - 12g + 6 - 19c = 0

Combining the constants, we get:

43 - 12g - 19c = 0

Now we have two equations involving 'g' and 'c':

1. g + 6c = 8 (from the center lying on the line)
2. 43 - 12g - 19c = 0 (from the point (6, 1) being on the circle)

We've got a system of two linear equations with two variables! This means we can solve for the exact values of 'g' and 'c'. Let's use the first equation to express 'g' in terms of 'c': g = 8 - 6c. Now, we can substitute this expression for 'g' into the second equation:

43 - 12(8 - 6c) - 19c = 0

Let's expand and simplify:

43 - 96 + 72c - 19c = 0

-53 + 53c = 0

From this, we can easily find 'c': 53c = 53, so c = 1.

Once we have 'c', we can find 'g' using g = 8 - 6c:

g = 8 - 6(1)

g = 8 - 6

g = 2.

So, we've found our values: g = 2 and c = 1. This is awesome because now we can write the specific equation of our circle and determine its radius.

Finding the Radius and X-intercept

With g = 2 and c = 1, our original circle equation x² + y² - 2gx + 6y - 19c = 0 becomes:

x² + y² - 2(2)x + 6y - 19(1) = 0

x² + y² - 4x + 6y - 19 = 0

To find the radius, let's convert this to the standard form (x - h)² + (y - k)² = r². We complete the square for the x-terms and y-terms:

(x² - 4x) + (y² + 6y) = 19

(x² - 4x + 4) + (y² + 6y + 9) = 19 + 4 + 9

(x - 2)² + (y + 3)² = 32

From this standard form, we can clearly see that the center of the circle is (h, k) = (2, -3), and the radius squared is r² = 32. So, the radius r = √32 = 4√2.

Now, let's talk about the X-axis intercept. The X-axis is defined by the equation y = 0. To find the points where the circle intersects the X-axis, we substitute y = 0 into the circle's equation:

x² + (0)² - 4x + 6(0) - 19 = 0

x² - 4x - 19 = 0

This is a quadratic equation for 'x'. The solutions to this equation, let's call them x₁ and x₂, represent the x-coordinates of the two points where the circle crosses the X-axis. The length of the intercept made by the circle on the X-axis is the distance between these two points, which is simply |x₁ - x₂|.

There's a neat little formula for the length of the X-intercept of a circle with center (h, k) and radius 'r'. If the circle intersects the X-axis, the points of intersection will have y-coordinate 0. So, we set y=0 in the standard equation: (x - h)² + (0 - k)² = r², which gives (x - h)² + k² = r². Rearranging, we get (x - h)² = r² - k². Taking the square root of both sides, x - h = ±√(r² - k²), so x = h ± √(r² - k²). The two x-intercepts are x₁ = h - √(r² - k²) and x₂ = h + √(r² - k²). The length of the intercept is |x₂ - x₁| = |(h + √(r² - k²)) - (h - √(r² - k²))| = |2√(r² - k²)|. This length is 2√(r² - k²), provided r² ≥ k² (meaning the circle actually intersects the X-axis).

In our case, the center is (2, -3), so h = 2 and k = -3. The radius squared is r² = 32.

The length of the X-intercept is 2√(r² - k²).

Plugging in our values:

**Length = 2√(32 - (-3)²) **

Length = 2√(32 - 9)

Length = 2√(23)

Wait a minute... that doesn't match any of the options. Let's double-check our calculations. Ah, I see a potential source of confusion. The general equation for the X-intercept length is indeed 2√(r² - k²). However, sometimes it's easier to work directly with the quadratic equation we got: x² - 4x - 19 = 0. For a quadratic equation ax² + bx + c = 0, the difference between the roots (x₁ and x₂) is given by |x₁ - x₂| = √(b² - 4ac) / |a|.

In our case, a = 1, b = -4, and c = -19.

So, the length of the intercept is √((-4)² - 4 * 1 * -19) / |1|

Length = √(16 + 76)

Length = √92

This still doesn't look right and it's not matching the options. Let's re-read the question and my steps very carefully. Is there a simpler way to interpret the X-intercept length directly from the general equation?

Revisiting the X-intercept Calculation

Okay, let's go back to basics. The general form of the circle is x² + y² - 2gx + 6y - 19c = 0. The length of the intercept on the X-axis is given by 2√(g² - ( -19c )), where 'g' is the x-coordinate of the center and '-19c' is the constant term related to the radius calculation on the x-axis. More precisely, if the circle is (x-h)² + (y-k)² = r², the X-intercept is 2√(r² - k²). We found our specific circle equation to be x² + y² - 4x + 6y - 19 = 0. Here, 2g = 4, so g = 2, and k = -3. The constant term is -19. The radius squared, r², can be found from r² = g² + k² - C, where C is the constant term in the general equation. So, r² = (2)² + (-3)² - (-19) = 4 + 9 + 19 = 32.

The length of the intercept on the X-axis is 2√(r² - k²).

**Length = 2√(32 - (-3)²) **

Length = 2√(32 - 9)

Length = 2√(23). Still √23. This indicates I might have made a calculation error earlier or there's a misunderstanding of the formula application.

Let's re-evaluate the relationship between the general form and the intercept. The general equation of a circle is $x^2 + y^2 + 2gx + 2fy + c = 0$. The center is $(-g, -f)$ and the radius is $r = \sqrt{g^2 + f^2 - c}$. The length of the intercept on the x-axis is $2\sqrt{g^2 - c}$ (provided $g^2 \geq c$).

Our given equation is $x^2 + y^2 - 2gx + 6y - 19c = 0$. Comparing this to the standard general form, we have:

• $2g_{std} = -2g ightarrow g_{std} = -g$
• $2f_{std} = 6 ightarrow f_{std} = 3$
• $c_{std} = -19c$

So, the center is $(-g_{std}, -f_{std}) = (g, -3)$. This matches what we found.

The radius squared is $r^2 = g_{std}^2 + f_{std}^2 - c_{std} = (-g)^2 + (3)^2 - (-19c) = g^2 + 9 + 19c$.

The length of the X-intercept is $2\sqrt{g_{std}^2 - c_{std}}$ (provided $g_{std}^2 \geq c_{std}$).

So, Length = $2\sqrt{(-g)^2 - (-19c)} = 2\sqrt{g^2 + 19c}$.

Now we need to use our values of g=2 and c=1.

Length = $2\sqrt{(2)^2 + 19(1)}$

Length = $2\sqrt{4 + 19}$

Length = 2√23. It seems I'm consistently getting this result. Let me check the formula for the X-intercept length once more. It should be $2 extrm{sqrt}(r^2 - k^2)$ where $r$ is the radius and $k$ is the y-coordinate of the center.

We had $r^2 = 32$ and $k = -3$. So $2 extrm{sqrt}(32 - (-3)^2) = 2 extrm{sqrt}(32 - 9) = 2 extrm{sqrt}(23)$.

Could there be a mistake in the problem statement or the options provided? Let's re-verify everything from the start, focusing on potential pitfalls.

We derived g = 2 and c = 1. This means the circle equation is x² + y² - 4x + 6y - 19 = 0.

To find the X-intercept, we set y = 0: x² - 4x - 19 = 0.

The roots of this quadratic equation are $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Here, $a=1, b=-4, c=-19$.

$x = \frac{4 \pm \sqrt{(-4)^2 - 4(1)(-19)}}{2(1)}$

$x = \frac{4 \pm \sqrt{16 + 76}}{2}$

$x = \frac{4 \pm \sqrt{92}}{2}$

$x = \frac{4 \pm 2\sqrt{23}}{2}$

$x = 2 \pm \sqrt{23}$

The two x-intercepts are $x_1 = 2 - \sqrt{23}$ and $x_2 = 2 + \sqrt{23}$.

The length of the intercept is $|x_2 - x_1| = |(2 + \sqrt{23}) - (2 - \sqrt{23})| = |2\sqrt{23}| = 2\sqrt{23}$.

It seems that my calculations are consistent, and the result $2 extrm{sqrt}(23)$ is what I am getting. This does not match any of the options (a) √11, (b) 4, (c) 3, (d) 2√3. This strongly suggests there might be an error in the question's provided options or the initial parameters.

However, let me consider if I interpreted