This problem is a comprehensive test of various concepts in coordinate geometry, specifically involving lines, circles, and ellipses. We'll systematically break down the problem into smaller, manageable parts.

Problem Overview: We are given a line that intersects the axes at points A and B. These points form the diameter of a circle, whose equation is provided. Our first goal is to use this information to find the unknown constant $k$. Once $k$ is known, we will substitute it into the equation of an ellipse. Finally, we need to find the length of the latus rectum of this ellipse and use it to calculate the value of $2m+n$.

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Step 1: Finding the Intercepts A and B of the Line

Concept: The $x$-intercept is the point where a line crosses the $x$-axis. At this point, the $y$-coordinate is $0$. Similarly, the $y$-intercept is the point where a line crosses the $y$-axis, meaning the $x$-coordinate is $0$.

Why this step? Points A and B are crucial because they define the diameter of the circle. To find their coordinates, we apply the fundamental definitions of $x$ and $y$-intercepts to the given line equation.

The given line equation is $2x+3y-k=0$.

• To find the $x$-intercept (Point A): We set $y=0$ in the line equation. $2x + 3(0) - k = 0$ $2x = k$ $x = \frac{k}{2}$ Therefore, point A, the $x$-intercept, is $\left(\frac{k}{2}, 0\right)$.

• To find the $y$-intercept (Point B): We set $x=0$ in the line equation. $2(0) + 3y - k = 0$ $3y = k$ $y = \frac{k}{3}$ Therefore, point B, the $y$-intercept, is $\left(0, \frac{k}{3}\right)$.

Tip: Always double-check your intercept calculations. A small error here will propagate through the entire problem and lead to an incorrect final answer.

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Step 2: Forming the Equation of the Circle with AB as Diameter

Concept: The equation of a circle whose diameter has endpoints $(x_1, y_1)$ and $(x_2, y_2)$ is given by the diameter form: $(x-x_1)(x-x_2) + (y-y_1)(y-y_2) = 0$ This formula is derived from the property that the angle subtended by a diameter at any point on the circumference is $90^\circ$.

Why this step? We have successfully found the coordinates of the two endpoints of the diameter, A and B. Using the diameter form of the circle equation allows us to directly construct the circle's equation based on these points, which we can then compare with the given circle equation to find $k$.

Let $(x_1, y_1) = \left(\frac{k}{2}, 0\right)$ (point A) and $(x_2, y_2) = \left(0, \frac{k}{3}\right)$ (point B). Substitute these coordinates into the diameter form of the circle equation: $\left(x - \frac{k}{2}\right)(x - 0) + (y - 0)\left(y - \frac{k}{3}\right) = 0$ Simplify the terms: $x\left(x - \frac{k}{2}\right) + y\left(y - \frac{k}{3}\right) = 0$ Expand the expressions: $x^2 - \frac{kx}{2} + y^2 - \frac{ky}{3} = 0$ Rearrange it into the general form $x^2+y^2+Gx+Fy+C=0$: $x^2 + y^2 - \frac{k}{2}x - \frac{k}{3}y = 0$ This is the equation of the circle derived from the line segment AB as its diameter.

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Step 3: Determining the Value of k by Comparing Circle Equations

Concept: If two algebraic equations represent the same geometric object (in this case, the same circle), then their corresponding coefficients must be equal. This principle holds true when the coefficients of $x^2$ and $y^2$ are normalized to 1, as they are in both equations.

Why this step? The problem provides us with the equation of the circle as $x^2+y^2-3x-2y=0$. We have just derived an alternative equation for the same circle. By comparing the coefficients of the corresponding terms (specifically $x$ and $y$) in both equations, we can solve for the unknown parameter $k$.

Our derived equation: $x^2 + y^2 - \frac{k}{2}x - \frac{k}{3}y = 0$ Given equation: $x^2 + y^2 - 3x - 2y = 0$

Comparing the coefficients of $x$: $-\frac{k}{2} = -3$ Multiply both sides by $-2$: $k = 6$

Comparing the coefficients of $y$: $-\frac{k}{3} = -2$ Multiply both sides by $-3$: $k = 6$

Both comparisons consistently yield $k=6$. The problem statement specifies that $k>0$, which our value $k=6$ satisfies.

Tip: Always compare coefficients for all relevant terms (e.g., $x$, $y$, and the constant term if present) to ensure consistency. If the values of $k$ obtained from different coefficient comparisons do not match, it indicates an error in an earlier step of your calculations.

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Step 4: Substituting k into the Ellipse Equation

Concept: Direct substitution is a fundamental algebraic operation where a known value for a variable is placed into an equation to simplify it or solve for other unknowns.

Why this step? Now that we have successfully determined the value of $k=6$, we can substitute it into the given equation of the ellipse. This will give us the specific numerical equation of the ellipse, which is a necessary prerequisite for finding its latus rectum.

The equation of the ellipse is given as $x^2+9y^2=k^2$. Substitute $k=6$ into the equation: $x^2+9y^2=(6)^2$ $x^2+9y^2=36$

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Step 5: Converting the Ellipse Equation to Standard Form

Concept: The standard form of an ellipse centered at the origin is $\frac{x^2}{a_{maj}^2} + \frac{y^2}{a_{min}^2} = 1$ or $\frac{x^2}{a_{min}^2} + \frac{y^2}{a_{maj}^2} = 1$. Here, $a_{maj}$ represents the length of the semi-major axis (half of the longest diameter), and $a_{min}$ represents the length of the semi-minor axis (half of the shortest diameter). The orientation of the major axis (along $x$ or $y$) depends on which denominator is larger.

Why this step? To accurately calculate the length of the latus rectum, we need to identify the precise lengths of the semi-major axis ($a_{maj}$) and the semi-minor axis ($a_{min}$). Converting the ellipse equation to its standard form makes these values immediately apparent and correctly distinguishes between $a_{maj}$ and $a_{min}$.

Our ellipse equation is $x^2+9y^2=36$. To achieve the standard form, we need the right-hand side of the equation to be equal to 1. We accomplish this by dividing the entire equation by 36: $\frac{x^2}{36} + \frac{9y^2}{36} = \frac{36}{36}$ Simplify the second term: $\frac{x^2}{36} + \frac{y^2}{4} = 1$

Now, we compare this with the standard form $\frac{x^2}{A^2} + \frac{y^2}{B^2} = 1$: We have $A^2 = 36 \Rightarrow A = \sqrt{36} = 6$. And $B^2 = 4 \Rightarrow B = \sqrt{4} = 2$.

Since $A^2=36$ is greater than $B^2=4$, the major axis of this ellipse lies along the $x$-axis. Therefore, the semi-major axis is $a_{maj} = A = 6$. The semi-minor axis is $a_{min} = B = 2$.

Common Mistake: It's a common misconception to always assign $a$ as the semi-major axis and $b$ as the semi-minor axis without checking their values. In the standard form $\frac{x^2}{A^2} + \frac{y^2}{B^2} = 1$, the semi-major axis is always the square root of the larger denominator, and the semi-minor axis is the square root of the smaller denominator. In this case, $A=6$ is indeed the semi-major axis and $B=2$ is the semi-minor axis.

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Step 6: Calculating the Length of the Latus Rectum

Concept: For an ellipse, the length of the latus rectum (LR) is a chord passing through a focus and perpendicular to the major axis. Its length is given by the formula: $LR = \frac{2(a_{min})^2}{a_{maj}}$ where $a_{maj}$ is the length of the semi-major axis and $a_{min}$ is the length of the semi-minor axis.

Why this step? This is the final geometric calculation explicitly requested by the problem statement before we can determine the values of $m$ and $n$.

Using the values determined in Step 5: $a_{maj} = 6$ $a_{min} = 2$

Substitute these values into the formula for the length of the latus rectum: $LR = \frac{2 \times (2)^2}{6}$ $LR = \frac{2 \times 4}{6}$ $LR = \frac{8}{6}$ Simplify the fraction by dividing both the numerator and denominator by their greatest common divisor, which is 2: $LR = \frac{4}{3}$

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Step 7: Determining m, n, and Calculating $2m+n$

Concept: Coprime integers (also known as relatively prime integers) are two integers whose greatest common divisor (GCD) is 1. This means they share no common positive factors other than 1.

Why this step? The problem states that the length of the latus rectum is $\frac{m}{n}$, where $m$ and $n$ are coprime. This requires us to express our calculated fraction in its simplest form and then identify the corresponding $m$ and $n$ values before performing the final arithmetic calculation.

We found the length of the latus rectum to be $LR = \frac{4}{3}$. Comparing this with the given form $\frac{m}{n}$: $m = 4$ $n = 3$

Now, we must verify if $m$ and $n$ are coprime. The factors of 4 are $\{1, 2, 4\}$. The factors of 3 are $\{1, 3\}$. The greatest common divisor of 4 and 3 is 1. Thus, 4 and 3 are indeed coprime.

Finally, we need to calculate the expression $2m+n$: $2m+n = 2(4) + 3$ $2m+n = 8 + 3$ $2m+n = 11$

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Summary and Key Takeaway:

This problem is an excellent example of how multiple concepts from coordinate geometry are integrated into a single question. A systematic approach is crucial for success:

1. Line Intercepts: Accurately finding the points where a line crosses the axes.
2. Circle Equation (Diameter Form): Effectively using the specific formula for a circle when the endpoints of its diameter are known. This often bypasses the need to find the center and radius separately.
3. Coefficient Comparison: A powerful algebraic technique to equate unknown parameters when two different equations represent the same geometric entity.
4. Ellipse Standard Form: Converting a given ellipse equation into its standard form is essential for correctly identifying its key properties, such as the lengths of the semi-major and semi-minor axes.
5. Latus Rectum of an Ellipse: Applying the correct formula for the length of the latus rectum, paying attention to which axis is the major and which is the minor.
6. Coprime Integers: Understanding the definition to correctly extract $m$ and $n$ from a fraction in its simplest form.

The final answer is $\boxed{\text{11}}$.