Key Concepts and Formulas for Parabola and Focal Chords

For a parabola with its vertex at the origin and axis along the x-axis, its standard equation is given by: $y^2 = 4ax$ Where:

1. Vertex: The point $(0, 0)$.
2. Focus: The point $(a, 0)$. This is a crucial point for defining a parabola.
3. Directrix: The line $x = -a$.
4. Parametric Coordinates: Any point $P$ on the parabola $y^2 = 4ax$ can be uniquely represented by a parameter $t$ as $P(at^2, 2at)$. This form is exceptionally useful for problems involving chords and tangents as it simplifies calculations from two variables $(x, y)$ to a single parameter $(t)$.

Focal Chord Properties: A focal chord is a chord of the parabola that passes through its focus. If $P(at_1^2, 2at_1)$ is one end of a focal chord, and $Q(at_2^2, 2at_2)$ is the other end, then a fundamental property relating their parameters is: $t_2 = -\frac{1}{t_1}$ Why this property holds: A line passing through two points $P(x_1, y_1)$ and $Q(x_2, y_2)$ and the focus $F(a, 0)$ implies that the points $P, Q, F$ are collinear. The slope of $PF$ must be equal to the slope of $QF$. Slope of $PF = \frac{2at_1 - 0}{at_1^2 - a} = \frac{2at_1}{a(t_1^2 - 1)} = \frac{2t_1}{t_1^2 - 1}$. Slope of $QF = \frac{2at_2 - 0}{at_2^2 - a} = \frac{2at_2}{a(t_2^2 - 1)} = \frac{2t_2}{t_2^2 - 1}$. Since $F$ lies on the chord $PQ$, the slope of $PQ$ must be equal to the slope of $PF$ (and $QF$). Slope of $PQ = \frac{2at_1 - 2at_2}{at_1^2 - at_2^2} = \frac{2a(t_1 - t_2)}{a(t_1 - t_2)(t_1 + t_2)} = \frac{2}{t_1 + t_2}$. Equating the slopes: $\frac{2}{t_1 + t_2} = \frac{2t_1}{t_1^2 - 1}$. This simplifies to $t_1^2 - 1 = t_1(t_1 + t_2) \implies t_1^2 - 1 = t_1^2 + t_1 t_2 \implies -1 = t_1 t_2 \implies t_2 = -\frac{1}{t_1}$.

Length of a Focal Chord: The distance between two points $P(at_1^2, 2at_1)$ and $Q(at_2^2, 2at_2)$ can be calculated using the distance formula. Length $L = \sqrt{(at_1^2 - at_2^2)^2 + (2at_1 - 2at_2)^2}$ $L = \sqrt{a^2(t_1^2 - t_2^2)^2 + 4a^2(t_1 - t_2)^2}$ $L = \sqrt{a^2(t_1 - t_2)^2(t_1 + t_2)^2 + 4a^2(t_1 - t_2)^2}$ $L = a|t_1 - t_2|\sqrt{(t_1 + t_2)^2 + 4}$ Now, substitute $t_2 = -1/t_1$: $t_1 - t_2 = t_1 - (-\frac{1}{t_1}) = t_1 + \frac{1}{t_1}$ $t_1 + t_2 = t_1 + (-\frac{1}{t_1}) = t_1 - \frac{1}{t_1}$ $L = a\left|t_1 + \frac{1}{t_1}\right|\sqrt{\left(t_1 - \frac{1}{t_1}\right)^2 + 4}$ We know that $(A-B)^2 + 4AB = (A+B)^2$. Here, $A=t_1$ and $B=1/t_1$, so $AB=1$. Thus, $\left(t_1 - \frac{1}{t_1}\right)^2 + 4 = \left(t_1 + \frac{1}{t_1}\right)^2$. Substituting this into the length formula: $L = a\left|t_1 + \frac{1}{t_1}\right|\sqrt{\left(t_1 + \frac{1}{t_1}\right)^2} = a\left|t_1 + \frac{1}{t_1}\right|\left|t_1 + \frac{1}{t_1}\right|$ Since $t_1 + \frac{1}{t_1}$ can be positive or negative, its square is always positive. The length itself must be positive. Therefore, the formula is: $L = a \left(t_1 + \frac{1}{t_1}\right)^2$ An alternative and often simpler form for calculation, if both $t_1$ and $t_2$ are known, is: $L = a(t_1 - t_2)^2$ Both formulas are equivalent and valid.

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Step-by-Step Solution

Our goal is to find the length of a focal chord given the parabola's equation and one of its endpoints. We will use the parametric form and properties of focal chords.

1. Identify the Parabola's Parameter 'a' The given equation of the parabola is $y^2 = 16x$. Why this step is taken: The parameter 'a' is fundamental to defining the specific parabola and its focus. All subsequent calculations involving parametric points and chord lengths depend on 'a'.

We compare this equation with the standard form of a parabola, $y^2 = 4ax$. By comparing the coefficients of $x$, we have: $4a = 16$ To find 'a', we divide both sides by 4: $a = \frac{16}{4} = 4$ This tells us that the focus of this parabola is at $(a, 0) = (4, 0)$.

2. Determine the Parameter $t_1$ for the Given End Point We are given that one end of the focal chord is at the point $P(1, 4)$. Why this step is taken: To utilize the parametric formulas for focal chords, we need to convert the given Cartesian coordinates of the point into its corresponding parameter $t$.

We know that any point on the parabola $y^2 = 4ax$ can be represented parametrically as $(at^2, 2at)$. So, for the point $P(1, 4)$, we equate its coordinates with $(at_1^2, 2at_1)$: $at_1^2 = 1 \quad \text{and} \quad 2at_1 = 4$ Now, substitute the value $a=4$ (found in Step 1) into these two equations: From the x-coordinate: $4t_1^2 = 1 \implies t_1^2 = \frac{1}{4}$ Taking the square root of both sides, we get $t_1 = \pm \frac{1}{2}$.

From the y-coordinate: $2(4)t_1 = 4 \implies 8t_1 = 4$ Dividing by 8, we get $t_1 = \frac{4}{8} = \frac{1}{2}$.

Both coordinate equations consistently yield $t_1 = \frac{1}{2}$. This confirms the parameter for the point $P(1, 4)$.

3. Determine the Parameter $t_2$ for the Other End Point Since $PQ$ is a focal chord, the parameters of its endpoints are related by the property $t_2 = -\frac{1}{t_1}$. Why this step is taken: This property is a direct consequence of the chord passing through the focus. It allows us to find the parameter of the second endpoint without needing to explicitly find its Cartesian coordinates, simplifying the process.

Using the value $t_1 = \frac{1}{2}$ that we just found: $t_2 = -\frac{1}{\frac{1}{2}}$ $t_2 = -2$ So, the parameter for the other end of the focal chord, $Q$, is $t_2 = -2$.

4. Calculate the Length of the Focal Chord Now we can use the formula for the length of a focal chord. We will use the formula $L = a(t_1 - t_2)^2$ as it directly uses the parameters $t_1$ and $t_2$ we have already found. Why this step is taken: This is the final step to answer the question, applying the derived formula with the calculated values of 'a', $t_1$, and $t_2$.

Substitute $a=4$, $t_1=\frac{1}{2}$, and $t_2=-2$ into the formula: $L = 4 \left(\frac{1}{2} - (-2)\right)^2$ First, simplify the expression inside the parenthesis: $L = 4 \left(\frac{1}{2} + 2\right)^2$ To add the terms inside the parenthesis, find a common denominator: $L = 4 \left(\frac{1}{2} + \frac{4}{2}\right)^2$ $L = 4 \left(\frac{1+4}{2}\right)^2$ $L = 4 \left(\frac{5}{2}\right)^2$ Next, square the term in the parenthesis: $L = 4 \left(\frac{25}{4}\right)$ Finally, multiply to get the length: $L = 25$

The length of the focal chord is 25 units.

Verification using an Alternative Formula (Optional but Recommended) Another formula for the length of a focal chord, if its endpoints are $(x_1, y_1)$ and $(x_2, y_2)$, is $L = x_1 + x_2 + 2a$. We have $x_1 = 1$ (given point is $(1,4)$) and $a=4$. We need to find $x_2$. The x-coordinate of the second endpoint $Q$ is $at_2^2$. $x_2 = a t_2^2 = 4(-2)^2 = 4(4) = 16$. Now, substitute these values into the alternative formula: $L = x_1 + x_2 + 2a = 1 + 16 + 2(4)$ $L = 17 + 8$ $L = 25$ This independent calculation confirms our previous result, giving us high confidence in the answer.

The calculated length is 25. Comparing this to the given options: (A) 24 (B) 20 (C) 25 (D) 22 Our calculated answer matches option (C). (Note: The problem statement indicates option (A) 24 as the correct answer. However, based on standard mathematical formulas and rigorous calculation, the length of the focal chord is 25. It is possible there is a discrepancy in the provided options or the intended correct answer.)

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Tips and Common Mistakes to Avoid

• Standard Forms are Key: Always begin by correctly identifying the standard form of the given conic section ($y^2 = 4ax$ in this case) and extracting its parameters, especially 'a'. A mistake here will propagate throughout the entire solution.
• Parametric Representation: For problems involving chords, especially focal chords, the parametric form $(at^2, 2at)$ is incredibly powerful and simplifies calculations. Resist the urge to work solely with Cartesian coordinates, as it can lead to more complex algebra.
• Focal Chord Property: Remember the crucial relationship $t_2 = -1/t_1$ for focal chords. A common mistake is forgetting the negative sign or confusing it with other chord properties.
• Arithmetic Accuracy: Be meticulous with arithmetic, especially when dealing with fractions, squaring, and signs. A small calculation error can lead to an incorrect final answer. Double-check your steps.
• Alternative Formulas for Verification: Whenever possible, use an alternative formula or method to verify your answer. This builds confidence and helps catch errors. For focal chords, $L = a(t_1 + 1/t_1)^2$ and $L = x_1 + x_2 + 2a$ are good cross-checks.

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Summary and Key Takeaway

This problem is a classic application of the parametric form of a parabola and the specific properties of a focal chord. The most efficient strategy involves:

1. Identifying the parameter 'a' from the parabola's equation.
2. Converting the given Cartesian coordinates of the endpoint into its parametric 't' value.
3. Applying the focal chord property ($t_2 =