Understanding the Parabola and its Key Properties

This problem tests our understanding of the fundamental properties of a parabola, particularly focusing on focal chords and focal distances. To solve it efficiently, we will rely on the following key concepts:

1. Standard Equation of a Parabola: The general equation of a parabola opening to the right with its vertex at the origin is $y^2 = 4ax$. The parameter '$a$' is crucial for defining all other properties.
2. Focus of the Parabola: For $y^2 = 4ax$, the focus is the point $S(a,0)$. This point is central to many parabola properties.
3. Parametric Coordinates: Any point P on the parabola $y^2 = 4ax$ can be represented parametrically as $P(at^2, 2at)$. This representation often simplifies calculations involving points on the parabola.
4. Focal Chord Property: If P and Q are the endpoints of a focal chord of the parabola $y^2 = 4ax$, and their respective parameters are $t_P$ and $t_Q$, then they are related by the fundamental property: $t_P \cdot t_Q = -1$ This means if one endpoint is $P(at^2, 2at)$, the other endpoint $Q$ will have parameter $-1/t$, so $Q(a/t^2, -2a/t)$.
5. Focal Distance: The distance of any point $(x,y)$ on the parabola $y^2 = 4ax$ from its focus $(a,0)$ is called its focal distance. It can be directly calculated as: $PS = x + a$ Alternatively, using the parametric form, $PS = at^2 + a = a(t^2+1)$. This formula significantly streamlines length calculations.
6. Section Formula (Ratio of Division): If a point S divides a line segment PQ in the ratio $m:n$, it means that the ratio of the lengths of the segments $PS$ to $SQ$ is $m:n$. That is, $PS:SQ = m:n$.

Let's now apply these concepts step-by-step to solve the problem.

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Step-by-Step Solution

Step 1: Identify Parabola Parameters and Focus

The first crucial step is to extract the fundamental parameter 'a' from the given parabola equation, as 'a' defines the parabola's geometry, including its focus.

We are given the equation of the parabola: $y^2 = 16x$ Why this step? By comparing this to the standard form $y^2 = 4ax$, we can determine the value of 'a', which is essential for finding the focus and using parametric coordinates.

Comparing $y^2 = 16x$ with $y^2 = 4ax$: $4a = 16$ Solving for 'a': $a = \frac{16}{4} = 4$ Now that we have 'a', we can determine the coordinates of the focus, $S$: $S = (a, 0) = (4, 0)$

Step 2: Determine the Parameter for Point P

We are given one endpoint of the focal chord, $P(1, -4)$. Our next goal is to find the parameter '$t_P$' corresponding to this point.

Why this step? Using the parametric form for P allows us to easily find the parameter for Q (the other end of the focal chord) using the focal chord property $t_P t_Q = -1$, which is much simpler than other methods.

The parametric coordinates for a point on the parabola $y^2 = 4ax$ are $(at^2, 2at)$. Substituting $a=4$, the parametric form for our specific parabola is $(4t^2, 8t)$. For point $P(1, -4)$, we equate its coordinates with the parametric form: $4t_P^2 = 1 \quad \text{and} \quad 8t_P = -4$ Solving the linear equation for $t_P$ is generally easier: $t_P = \frac{-4}{8} = -\frac{1}{2}$ We should always verify this value with the other equation to ensure consistency: $4\left(-\frac{1}{2}\right)^2 = 4\left(\frac{1}{4}\right) = 1$ Both equations are satisfied, so the parameter for point P is $t_P = -\frac{1}{2}$.

Step 3: Determine the Parameter and Coordinates for Point Q

Since PQ is a focal chord, we can use the powerful focal chord property to find the parameter of the other endpoint, Q.

Why this step? The property $t_P \cdot t_Q = -1$ is a direct and efficient way to find the parameter of Q. This avoids the lengthier process of finding the equation of the line passing through P and S, and then finding its intersection with the parabola.

We found $t_P = -\frac{1}{2}$. Using the focal chord property $t_P \cdot t_Q = -1$: $\left(-\frac{1}{2}\right) \cdot t_Q = -1$ Multiplying both sides by -2: $t_Q = 2$ Now that we have the parameter $t_Q = 2$ and $a=4$, we can find the coordinates of point Q using the parametric form $(at^2, 2at)$: $Q(at_Q^2, 2at_Q) = Q(4(2^2), 2(4)(2))$ $Q(4 \cdot 4, 16) = Q(16, 16)$ Thus, the coordinates of the other endpoint of the focal chord are $Q(16, 16)$.

Step 4: Calculate Focal Distances PS and QS

The problem asks for the ratio in which the focus S divides the chord PQ. This means we need to find the lengths of the segments PS and QS.

Why this step? The ratio of division is defined by the ratio of the lengths of the segments from the dividing point (focus S) to the endpoints of the chord (P and Q). The focal distance formula provides an elegant and efficient way to calculate these lengths without resorting to the general distance formula.

We will use the focal distance formula: for any point $(x,y)$ on the parabola $y^2=4ax$, its distance from the focus $(a,0)$ is $x+a$.

For point $P(1, -4)$: The x-coordinate of P is $x_P = 1$. $PS = x_P + a = 1 + 4 = 5$ Alternatively, using the parametric form: $PS = a(t_P^2+1) = 4\left(\left(-\frac{1}{2}\right)^2+1\right) = 4\left(\frac{1}{4}+1\right) = 4\left(\frac{5}{4}\right) = 5$.

For point $Q(16, 16)$: The x-coordinate of Q is $x_Q = 16$. $QS = x_Q + a = 16 + 4 = 20$ Alternatively, using the parametric form: $QS = a(t_Q^2+1) = 4(2^2+1) = 4(4+1) = 4(5) = 20$.

Step 5: Determine the Ratio $m:n$

The focus S divides the chord PQ in the ratio $PS:SQ$. We need to express this ratio in its simplest form where $m$ and $n$ are coprime ($\gcd(m,n)=1$).

Why this step? We have calculated the individual lengths PS and QS. Now we form their ratio and simplify it to satisfy the problem's condition for $m$ and $n$.

The ratio is: $PS:QS = 5:20$ To express this ratio in its simplest form, we divide both numbers by their greatest common divisor (GCD), which is 5: $PS:QS = \frac{5}{5}:\frac{20}{5} = 1:4$ We are given that this ratio is $m:n$ and $\gcd(m,n)=1$. Therefore, by comparing $1:4$ with $m:n$, we get $m=1$ and $n=4$. Indeed, $\gcd(1,4)=1$, so these values are correct.

Step 6: Calculate $m^2+n^2$

Finally, we calculate the required expression $m^2+n^2$ using the values of $m$ and $n$ we found.

Why this step? This is the final calculation required by the problem statement.

Substitute $m=1$ and $n=4$ into the expression: $m^2+n^2 = 1^2 + 4^2 = 1 + 16 = 17$

Thus, the value of $m^2+n^2$ is 17.

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Tips for Success & Common Pitfalls

• Always identify 'a' first: The parameter 'a' is the fundamental building block for understanding all properties of the parabola $y^2=4ax$. Ensure you extract it correctly from the given equation.
• Master Parametric Form: Parametric coordinates $(at^2, 2at)$ are incredibly powerful for solving parabola problems, especially those involving chords, tangents, and normals. They often simplify complex algebraic manipulations significantly.
• Focal Chord Property is Key: The relation $t_P t_Q = -1$ for the parameters of endpoints of a focal chord is a frequent and crucial shortcut. Memorize and understand its application.
• Focal Distance Shortcut: The formula for focal distance ($x+a$ or $a(t^2+1)$) saves significant time compared to using the standard distance formula between two points. Use it whenever applicable.
• Simplify Ratios Correctly: Always ensure your ratio $m:n$ is in its simplest form (i.e., $\gcd(m,n)=1$) as specified in the problem. Failing to simplify will lead to incorrect $m$ and $n$ values, and subsequently, an incorrect final answer.
• Understand the Ratio Order: The problem states "focus divides the chord PQ in the ratio m:n", which implies the ratio of segment lengths $PS:SQ = m:n$. Be careful not to reverse the order (e.g., $QS:PS$), as this would swap $m$ and $n$.

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Summary and Key Takeaway

This problem is an excellent demonstration of how efficiently parabola problems can be solved by strategically leveraging the specific properties of conic sections. By systematically:

1. Identifying the parabola's fundamental parameter 'a' and its focus.
2. Utilizing the parametric form to represent points on the parabola and determine their 't' values.
3. Applying the focal chord property ($t_P t_Q = -1$) to find the other endpoint.
4. Using the focal distance formula ($x+a$) to quickly calculate segment lengths.
5. Simplifying the resulting ratio to meet the $\gcd(m,n)=1$ condition.

This approach avoids cumbersome coordinate geometry calculations and highlights the elegance and power of using these specialized properties. The final answer is $\boxed{\text{17}}$.