Question Statement

Prove that the semi latus rectum of a parabola is the harmonic mean between the segments of any focal chord.

Background and Explanation

To solve this problem, we use the standard equation of a parabola $y^{2} = 4 a x$ , where the focus $F$ is located at $(a, 0)$ . A focal chord is a line segment passing through the focus with endpoints on the parabola; for any such chord with parametric endpoints $t_{1}$ and $t_{2}$ , the property $t_{1} t_{2} = - 1$ always holds.

Solution

Consider the standard parabola $y^{2} = 4 a x$ with focus $F (a, 0)$ . Let the endpoints of the focal chord be $P (a t_{1}^{2}, 2 a t_{1})$ and $Q (a t_{2}^{2}, 2 a t_{2})$ . Since $P Q$ is a focal chord, the parameters $t_{1}$ and $t_{2}$ satisfy the relation: $t_{1} t_{2} = - 1$

The segments of the focal chord are the distances $∣ P F ∣$ and $∣ F Q ∣$ .

Step 1: Calculate the length of segment $∣ P F ∣$

Using the distance formula between $P (a t_{1}^{2}, 2 a t_{1})$ and $F (a, 0)$ : $∣ P F ∣ ∣ P F ∣ = (a t_{1}^{2} - a)^{2} + (2 a t_{1} - 0)^{2} = a^{2} (t_{1}^{2} - 1)^{2} + 4 a^{2} t_{1}^{2} = a (t_{1}^{2} - 1)^{2} + 4 t_{1}^{2} = a t_{1}^{4} - 2 t_{1}^{2} + 1 + 4 t_{1}^{2} = a t_{1}^{4} + 2 t_{1}^{2} + 1 = a (t_{1}^{2} + 1)^{2} = a (t_{1}^{2} + 1)$

Step 2: Calculate the length of segment $∣ F Q ∣$

Similarly, for the point $Q (a t_{2}^{2}, 2 a t_{2})$ : $∣ F Q ∣ ∣ F Q ∣ = (a t_{2}^{2} - a)^{2} + (2 a t_{2} - 0)^{2} = a^{2} (t_{2}^{2} - 1)^{2} + 4 a^{2} t_{2}^{2} = a (t_{2}^{2} - 1)^{2} + 4 t_{2}^{2} = a t_{2}^{4} - 2 t_{2}^{2} + 1 + 4 t_{2}^{2} = a t_{2}^{4} + 2 t_{2}^{2} + 1 = a (t_{2}^{2} + 1)^{2} = a (t_{2}^{2} + 1)$

Step 3: Calculate the Harmonic Mean of the segments

The harmonic mean (HM) of two values $x$ and $y$ is given by $\frac{2 x y}{x + y}$ . Let $x = ∣ P F ∣$ and $y = ∣ F Q ∣$ : $H M = \frac{2∣ P F ∣∣ F Q ∣}{∣ P F ∣ + ∣ F Q ∣} = \frac{2 [ a ( t _{1}^{2} + 1 ) ] [ a ( t _{2}^{2} + 1 ) ]}{a ( t _{1}^{2} + 1 ) + a ( t _{2}^{2} + 1 )} = \frac{2 a ^{2} ( t _{1}^{2} t _{2}^{2} + t _{1}^{2} + t _{2}^{2} + 1 )}{a ( t _{1}^{2} + 1 + t _{2}^{2} + 1 )} = 2 a \frac{( t _{1} t _{2} ) ^{2} + t _{1}^{2} + t _{2}^{2} + 1}{t _{1}^{2} + t _{2}^{2} + 2}$

Since $t_{1} t_{2} = - 1$ , then $(t_{1} t_{2})^{2} = 1$ . Substituting this into the equation: $H M = 2 a \frac{1 + t _{1}^{2} + t _{2}^{2} + 1}{t _{1}^{2} + t _{2}^{2} + 2} = 2 a \frac{t _{1}^{2} + t _{2}^{2} + 2}{t _{1}^{2} + t _{2}^{2} + 2} = 2 a$

The length of the latus rectum of the parabola $y^{2} = 4 a x$ is $4 a$ . Therefore, $2 a$ is exactly half of the latus rectum, known as the semi latus rectum.

Thus, the semi latus rectum is the harmonic mean between the segments of the focal chord.

Key Formulas or Methods Used

Standard Parabola: $y^{2} = 4 a x$
Parametric Coordinates: $P (a t^{2}, 2 a t)$
Focal Chord Property: For a focal chord with parameters $t_{1}$ and $t_{2}$ , $t_{1} t_{2} = - 1$ .
Distance Formula: $d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$
Harmonic Mean: $H M = \frac{2 x y}{x + y}$
Semi Latus Rectum: $l = \frac{1}{2} (4 a) = 2 a$

Summary of Steps

Represent the endpoints of the focal chord using parametric coordinates $(a t^{2}, 2 a t)$ .
Use the property that for any focal chord, the product of the parameters $t_{1} t_{2} = - 1$ .
Calculate the lengths of the segments $∣ P F ∣$ and $∣ F Q ∣$ using the distance formula.
Substitute these lengths into the harmonic mean formula.
Simplify the resulting expression using the $t_{1} t_{2} = - 1$ property to show the result is $2 a$ .

Background and Explanation

To solve this problem, we use the standard equation of a parabola

y^{2} = 4 a x

, where the focus

F

is located at

(a, 0)

. A focal chord is a line segment passing through the focus with endpoints on the parabola; for any such chord with parametric endpoints

t_{1}

and

t_{2}

, the property

t_{1} t_{2} = - 1

always holds.

Solution

Consider the standard parabola

y^{2} = 4 a x

with focus

F (a, 0)

. Let the endpoints of the focal chord be

P (a t_{1}^{2}, 2 a t_{1})

and

Q (a t_{2}^{2}, 2 a t_{2})

. Since

P Q

is a focal chord, the parameters

t_{1}

and

t_{2}

satisfy the relation:

t_{1} t_{2} = - 1

The segments of the focal chord are the distances

∣ P F ∣

and

∣ F Q ∣

Using the distance formula between

P (a t_{1}^{2}, 2 a t_{1})

and

F (a, 0)

∣ P F ∣ ∣ P F ∣ = (a t_{1}^{2} - a)^{2} + (2 a t_{1} - 0)^{2} = a^{2} (t_{1}^{2} - 1)^{2} + 4 a^{2} t_{1}^{2} = a (t_{1}^{2} - 1)^{2} + 4 t_{1}^{2} = a t_{1}^{4} - 2 t_{1}^{2} + 1 + 4 t_{1}^{2} = a t_{1}^{4} + 2 t_{1}^{2} + 1 = a (t_{1}^{2} + 1)^{2} = a (t_{1}^{2} + 1)

Similarly, for the point

Q (a t_{2}^{2}, 2 a t_{2})

∣ F Q ∣ ∣ F Q ∣ = (a t_{2}^{2} - a)^{2} + (2 a t_{2} - 0)^{2} = a^{2} (t_{2}^{2} - 1)^{2} + 4 a^{2} t_{2}^{2} = a (t_{2}^{2} - 1)^{2} + 4 t_{2}^{2} = a t_{2}^{4} - 2 t_{2}^{2} + 1 + 4 t_{2}^{2} = a t_{2}^{4} + 2 t_{2}^{2} + 1 = a (t_{2}^{2} + 1)^{2} = a (t_{2}^{2} + 1)

The harmonic mean (HM) of two values

x

and

y

is given by

\frac{2 x y}{x + y}

. Let

x = ∣ P F ∣

and

y = ∣ F Q ∣

H M = \frac{2∣ P F ∣∣ F Q ∣}{∣ P F ∣ + ∣ F Q ∣} = \frac{2 [ a ( t _{1}^{2} + 1 ) ] [ a ( t _{2}^{2} + 1 ) ]}{a ( t _{1}^{2} + 1 ) + a ( t _{2}^{2} + 1 )} = \frac{2 a ^{2} ( t _{1}^{2} t _{2}^{2} + t _{1}^{2} + t _{2}^{2} + 1 )}{a ( t _{1}^{2} + 1 + t _{2}^{2} + 1 )} = 2 a \frac{( t _{1} t _{2} ) ^{2} + t _{1}^{2} + t _{2}^{2} + 1}{t _{1}^{2} + t _{2}^{2} + 2}

Since

t_{1} t_{2} = - 1

, then

(t_{1} t_{2})^{2} = 1

. Substituting this into the equation:

H M = 2 a \frac{1 + t _{1}^{2} + t _{2}^{2} + 1}{t _{1}^{2} + t _{2}^{2} + 2} = 2 a \frac{t _{1}^{2} + t _{2}^{2} + 2}{t _{1}^{2} + t _{2}^{2} + 2} = 2 a

The length of the latus rectum of the parabola

y^{2} = 4 a x

4 a

. Therefore,

2 a

is exactly half of the latus rectum, known as the semi latus rectum.

Thus, the semi latus rectum is the harmonic mean between the segments of the focal chord.

Summary of Steps

Represent the endpoints of the focal chord using parametric coordinates

(a t^{2}, 2 a t)

Use the property that for any focal chord, the product of the parameters

t_{1} t_{2} = - 1

Calculate the lengths of the segments

∣ P F ∣

and

∣ F Q ∣

using the distance formula.

Substitute these lengths into the harmonic mean formula.

Simplify the resulting expression using the

t_{1} t_{2} = - 1

property to show the result is

2 a

7.5 Q-12

Question Statement

Background and Explanation

Solution

Step 1: Calculate the length of segment $∣ P F ∣$

Step 2: Calculate the length of segment $∣ F Q ∣$

Step 3: Calculate the Harmonic Mean of the segments

Key Formulas or Methods Used

Summary of Steps

7.5 Q-12

Question Statement

Background and Explanation

Solution

Step 1: Calculate the length of segment $∣ P F ∣$

Step 2: Calculate the length of segment $∣ F Q ∣$

Step 3: Calculate the Harmonic Mean of the segments

Key Formulas or Methods Used

Summary of Steps

Question Statement

Background and Explanation

Solution

Step 1: Calculate the length of segment ∣PF∣

Step 2: Calculate the length of segment ∣FQ∣

Step 3: Calculate the Harmonic Mean of the segments

Key Formulas or Methods Used

Summary of Steps

Question Statement

Background and Explanation

Solution

Step 1: Calculate the length of segment ∣PF∣

Step 2: Calculate the length of segment ∣FQ∣

Step 3: Calculate the Harmonic Mean of the segments

Key Formulas or Methods Used

Summary of Steps

Step 1: Calculate the length of segment $∣ P F ∣$

Step 2: Calculate the length of segment $∣ F Q ∣$

Step 1: Calculate the length of segment $∣ P F ∣$

Step 2: Calculate the length of segment $∣ F Q ∣$