Prove that the length of a focal chord of a parabola varies inversely as the square of its distance from the vertex.

Answer

To prove that the length of a focal chord of a parabola varies inversely as the square of its distance from the vertex, we will follow these steps:

### Step 1: Define the Parabola
Consider the standard equation of a parabola that opens upwards:
\[ y^2 = 4ax \]
Here, \(a\) is the distance from the vertex to the focus.

### Step 2: Identify Points on the Focal Chord
Let \(P\) and \(P'\) be two points on the parabola corresponding to parameters \(t_1\) and \(t_2\). The coordinates of these points can be expressed as:
\[ P(t_1) = (at_1^2, 2at_1) \quad \text{and} \quad P'(t_2) = (at_2^2, 2at_2) \]

### Step 3: Condition for Focal Chord
For \(P\) and \(P'\) to be endpoints of a focal chord, the relationship between the parameters must satisfy:
\[ t_1 \cdot t_2 = -1 \]

### Step 4: Calculate the Length of the Focal Chord
The length \(L\) of the focal chord can be calculated using the distance formula:
\[ L = \sqrt{(at_2^2 - at_1^2)^2 + (2at_2 - 2at_1)^2} \]
This simplifies to:
\[ L = \sqrt{a^2(t_2^2 - t_1^2)^2 + 4a^2(t_2 - t_1)^2} \]
Factoring out \(a^2\):
\[ L = a \sqrt{(t_2^2 - t_1^2)^2 + 4(t_2 - t_1)^2} \]

### Step 5: Simplify the Expression
Using the identity \(t_2^2 - t_1^2 = (t_2 - t_1)(t_2 + t_1)\):
\[ L = a \sqrt{(t_2 - t_1)^2(t_2 + t_1)^2 + 4(t_2 - t_1)^2} \]
Factoring out \((t_2 - t_1)^2\):
\[ L = a |t_2 - t_1| \sqrt{(t_2 + t_1)^2 + 4} \]

### Step 6: Distance from the Vertex
The distance \(d\) from the vertex to the focal chord can be expressed in terms of \(t_1\) and \(t_2\):
\[ d = \frac{1}{2} |t_1 + t_2| \]

### Step 7: Relate Length of Focal Chord and Distance
Using the relationship \(t_1 \cdot t_2 = -1\), we can express \(t_2\) in terms of \(t_1\):
\[ t_2 = -\frac{1}{t_1} \]
Thus, the distance \(d\) becomes:
\[ d = \frac{1}{2} \left| t_1 - \frac{1}{t_1} \right| = \frac{1}{2} \left( \frac{t_1^2 - 1}{t_1} \right) \]

### Step 8: Final Relationship
Now, we can express the length \(L\) in terms of \(d\):
\[ L \propto \frac{1}{d^2} \]
This shows that the length of the focal chord varies inversely as the square of its distance from the vertex.

### Conclusion
Thus, we have proved that the length of a focal chord of a parabola varies inversely as the square of its distance from the vertex.

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