Question Statement

Prove that the tangent to the parabola at any point $P$ on the parabola makes equal angles with the line joining $P$ and its focus, and the line through $P$ parallel to the axis of the parabola. (This is known as the Reflecting property of a parabola).

Background and Explanation

To solve this problem, we use the coordinate geometry of a standard parabola $y^{2} = 4 a x$ . We represent a general point $P$ using parametric coordinates and use the slopes of the tangent line, the focal chord, and the horizontal line to calculate the angles between them using the formula $tan θ = \frac{m _{2} - m _{1}}{1 + m _{1} m _{2}}$ .

Solution

Consider the standard parabola $y^{2} = 4 a x$ . Let $P (a t^{2}, 2 a t)$ be any point on the parabola, and let $F (a, 0)$ be the focus.

1. Finding the slope of the tangent ( $m_{1}$ )

To find the slope of the tangent at point $P$ , we differentiate the equation of the parabola with respect to $x$ : $2 y \frac{d y}{d x} = 4 a$ $\Rightarrow \frac{d y}{d x} = \frac{2 a}{y}$

At the point $P (a t^{2}, 2 a t)$ , the slope $m_{1}$ is: $\frac{d y}{d x} = \frac{2 a}{2 a t} = \frac{1}{t}$ $\Rightarrow m_{1} = \frac{1}{t}$

2. Finding the slope of the line $\overline{P F}$ ( $m_{2}$ )

The line $\overline{P F}$ joins the point $P (a t^{2}, 2 a t)$ and the focus $F (a, 0)$ . Using the slope formula $m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$ : $Slope of \overline{P F} = \frac{2 a t - 0}{a t ^{2} - a}$ $= \frac{2 a t}{a ( t ^{2} - 1 )}$ $\Rightarrow m_{2} = \frac{2 t}{t ^{2} - 1}$

3. Finding the slope of the line parallel to the axis ( $m_{3}$ )

Let $ℓ$ be the line passing through $P$ and parallel to the axis of the parabola. Since the axis of the parabola $y^{2} = 4 a x$ is the $x$ -axis, the slope of line $ℓ$ is equal to the slope of the $x$ -axis: $\Rightarrow m_{3} = 0$

4. Calculating the angle between the tangent and $\overline{P F}$ ( $α$ )

Let $α$ be the angle between the tangent (slope $m_{1}$ ) and the line $\overline{P F}$ (slope $m_{2}$ ): $tan α = \frac{m _{2} - m _{1}}{1 + m _{2} m _{1}}$

Substituting the values: $tan α = \frac{\frac{2 t}{t ^{2} - 1} - \frac{1}{t}}{1 + ( \frac{2 t}{t ^{2} - 1} ) ( \frac{1}{t} )}$ $= \frac{\frac{2 t ^{2} - ( t ^{2} - 1 )}{t ( t ^{2} - 1 )}}{\frac{t ( t ^{2} - 1 ) + 2 t}{t ( t ^{2} - 1 )}}$ $= \frac{2 t ^{2} - t ^{2} + 1}{t ^{3} - t + 2 t}$ $= \frac{t ^{2} + 1}{t ^{3} + t} = \frac{t ^{2} + 1}{t ( t ^{2} + 1 )}$ $\Rightarrow tan α = \frac{1}{t}$

5. Calculating the angle between the tangent and line $ℓ$ ( $β$ )

Let $β$ be the angle between the tangent (slope $m_{1}$ ) and the line $ℓ$ (slope $m_{3}$ ): $tan β = \frac{m _{1} - m _{3}}{1 + m _{1} m _{3}}$

Substituting the values: $tan β = \frac{\frac{1}{t} - 0}{1 + ( \frac{1}{t} ) ( 0 )}$ $= \frac{\frac{1}{t}}{1} = \frac{1}{t}$ $\Rightarrow tan β = \frac{1}{t}$

Conclusion

Since $tan α = \frac{1}{t}$ and $tan β = \frac{1}{t}$ , it follows that: $tan α = tan β \Rightarrow α = β$ Thus, the tangent makes equal angles with the focal radius and the line parallel to the axis.

Key Formulas or Methods Used

Parametric Coordinates of Parabola: $P (a t^{2}, 2 a t)$
Slope of Tangent via Differentiation: $\frac{d y}{d x} = \frac{2 a}{y}$
Slope Formula: $m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$
Angle Between Two Lines: $tan θ = \frac{m _{2} - m _{1}}{1 + m _{1} m _{2}}$

Summary of Steps

Define the parabola $y^{2} = 4 a x$ and the coordinates of point $P (a t^{2}, 2 a t)$ and focus $F (a, 0)$ .
Differentiate the parabola equation to find the slope of the tangent ( $m_{1} = 1/ t$ ).
Calculate the slope of the line segment $P F$ ( $m_{2}$ ) using the two-point formula.
Identify the slope of the line parallel to the axis ( $m_{3} = 0$ ).
Use the tangent subtraction formula to find $tan α$ (angle between tangent and $P F$ ).
Use the same formula to find $tan β$ (angle between tangent and the horizontal line).
Show that $tan α = tan β$ , proving the angles are equal.

Background and Explanation

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

y^{2} = 4 a x

. We represent a general point

P

using parametric coordinates and use the slopes of the tangent line, the focal chord, and the horizontal line to calculate the angles between them using the formula

tan θ = \frac{m _{2} - m _{1}}{1 + m _{1} m _{2}}

Solution

Consider the standard parabola

y^{2} = 4 a x

. Let

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

be any point on the parabola, and let

F (a, 0)

be the focus.

To find the slope of the tangent at point

P

, we differentiate the equation of the parabola with respect to

x

2 y \frac{d y}{d x} = 4 a

\Rightarrow \frac{d y}{d x} = \frac{2 a}{y}

At the point

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

, the slope

m_{1}

is:

\frac{d y}{d x} = \frac{2 a}{2 a t} = \frac{1}{t}

\Rightarrow m_{1} = \frac{1}{t}

The line

\overline{P F}

joins the point

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

and the focus

F (a, 0)

. Using the slope formula

m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

Slope of \overline{P F} = \frac{2 a t - 0}{a t ^{2} - a}

= \frac{2 a t}{a ( t ^{2} - 1 )}

\Rightarrow m_{2} = \frac{2 t}{t ^{2} - 1}

Let

ℓ

be the line passing through

P

and parallel to the axis of the parabola. Since the axis of the parabola

y^{2} = 4 a x

is the

x

-axis, the slope of line

ℓ

is equal to the slope of the

x

-axis:

\Rightarrow m_{3} = 0

Let

α

be the angle between the tangent (slope

m_{1}

) and the line

\overline{P F}

(slope

m_{2}

tan α = \frac{m _{2} - m _{1}}{1 + m _{2} m _{1}}

Substituting the values:

tan α = \frac{\frac{2 t}{t ^{2} - 1} - \frac{1}{t}}{1 + ( \frac{2 t}{t ^{2} - 1} ) ( \frac{1}{t} )}

= \frac{\frac{2 t ^{2} - ( t ^{2} - 1 )}{t ( t ^{2} - 1 )}}{\frac{t ( t ^{2} - 1 ) + 2 t}{t ( t ^{2} - 1 )}}

= \frac{2 t ^{2} - t ^{2} + 1}{t ^{3} - t + 2 t}

= \frac{t ^{2} + 1}{t ^{3} + t} = \frac{t ^{2} + 1}{t ( t ^{2} + 1 )}

\Rightarrow tan α = \frac{1}{t}

Let

β

be the angle between the tangent (slope

m_{1}

) and the line

ℓ

(slope

m_{3}

tan β = \frac{m _{1} - m _{3}}{1 + m _{1} m _{3}}

Substituting the values:

tan β = \frac{\frac{1}{t} - 0}{1 + ( \frac{1}{t} ) ( 0 )}

= \frac{\frac{1}{t}}{1} = \frac{1}{t}

\Rightarrow tan β = \frac{1}{t}

Since

tan α = \frac{1}{t}

and

tan β = \frac{1}{t}

, it follows that:

tan α = tan β \Rightarrow α = β

Thus, the tangent makes equal angles with the focal radius and the line parallel to the axis.

Summary of Steps

Define the parabola

y^{2} = 4 a x

and the coordinates of point

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

and focus

F (a, 0)

Differentiate the parabola equation to find the slope of the tangent (

m_{1} = 1/ t

Calculate the slope of the line segment

P F

(

m_{2}

) using the two-point formula.

Identify the slope of the line parallel to the axis (

m_{3} = 0

Use the tangent subtraction formula to find

tan α

(angle between tangent and

P F

Use the same formula to find

tan β

(angle between tangent and the horizontal line).

Show that

tan α = tan β

, proving the angles are equal.

7.5 Q-11

Question Statement

Background and Explanation

Solution

1. Finding the slope of the tangent ( $m_{1}$ )

2. Finding the slope of the line $\overline{P F}$ ( $m_{2}$ )

3. Finding the slope of the line parallel to the axis ( $m_{3}$ )

4. Calculating the angle between the tangent and $\overline{P F}$ ( $α$ )

5. Calculating the angle between the tangent and line $ℓ$ ( $β$ )

Conclusion

Key Formulas or Methods Used

Summary of Steps

7.5 Q-11

Question Statement

Background and Explanation

Solution

1. Finding the slope of the tangent ( $m_{1}$ )

2. Finding the slope of the line $\overline{P F}$ ( $m_{2}$ )

3. Finding the slope of the line parallel to the axis ( $m_{3}$ )

4. Calculating the angle between the tangent and $\overline{P F}$ ( $α$ )

5. Calculating the angle between the tangent and line $ℓ$ ( $β$ )

Conclusion

Key Formulas or Methods Used

Summary of Steps