6.4 Q-9

Question Statement

Prove that the tangent at any point $P$ on a parabola makes equal angles with:

1. The line joining $P$ to the focus (angle of incidence).
2. The line through $P$ parallel to the axis of the parabola (angle of reflection).

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Background and Explanation

This problem leverages the reflective property of parabolas, where the tangent at any point reflects light or paths such that the angle of incidence equals the angle of reflection.

For a parabola $y^2 = 4ax$:

1. The axis of symmetry is the x-axis.
2. The focus is at $(a, 0)$.
3. The directrix is $x = -a$.

Key properties include:

• The slope of the tangent at a point can be derived by differentiating the equation of the parabola.
• The angles are related to the slopes of the tangent and the relevant lines.

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Solution

Step 1: Define the Parabola and Tangent

The equation of the parabola is:

$$y^2 = 4ax$$

Let $P(x_1, y_1)$ be a point on the parabola. Differentiating the parabola’s equation with respect to $x$:

$$2y \frac{dy}{dx} = 4a$$ $$\frac{dy}{dx} = \frac{2a}{y}$$

Thus, the slope of the tangent at $P(x_1, y_1)$ is:

$$m_1 = \frac{2a}{y_1}$$

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Step 2: Slope of the Line Through $P$ and Focus $F$

The focus of the parabola is $F(a, 0)$. The slope of the line joining $P(x_1, y_1)$ to $F(a, 0)$ is:

$$m_3 = \frac{y_1 - 0}{x_1 - a} = \frac{y_1}{x_1 - a}$$

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Step 3: Slope of the Line Parallel to the Axis

The axis of the parabola is the x-axis. A line through $P(x_1, y_1)$ parallel to the axis has slope:

$$m_2 = 0$$

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Step 4: Angle Between Lines

The angle between two lines with slopes $m_1$ and $m_3$ is given by:

$$\tan \theta_1 = \left|\frac{m_1 - m_3}{1 + m_1 m_3}\right|$$

Substituting $m_1 = \frac{2a}{y_1}$ and $m_3 = \frac{y_1}{x_1 - a}$:

$$\tan \theta_1 = \left|\frac{\frac{2a}{y_1} - \frac{y_1}{x_1 - a}}{1 + \left(\frac{2a}{y_1}\right)\left(\frac{y_1}{x_1 - a}\right)}\right|$$

After simplification:

$$\tan \theta_1 = \frac{-2a}{y_1}$$

For the angle between the tangent and the line parallel to the axis, where $m_1 = \frac{2a}{y_1}$ and $m_2 = 0$:

$$\tan \theta_2 = \left|\frac{m_2 - m_1}{1 + m_2 m_1}\right| = \left|\frac{0 - \frac{2a}{y_1}}{1 + 0}\right|$$ $$\tan \theta_2 = \frac{-2a}{y_1}$$

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Step 5: Conclusion

Since $\tan \theta_1 = \tan \theta_2$, the angles $\theta_1$ and $\theta_2$ are equal:

$$\theta_1 = \theta_2$$

Thus, the tangent at $P$ makes equal angles with the line joining $P$ to the focus and the line through $P$ parallel to the axis of the parabola.

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Key Formulas or Methods Used

1. Slope of Tangent:

$$m_1 = \frac{2a}{y_1}$$

2. Slope of Line Joining Focus to $P$:

$$m_3 = \frac{y_1}{x_1 - a}$$

3. Angle Between Two Lines:

$$\tan \theta = \left|\frac{m_1 - m_2}{1 + m_1 m_2}\right|$$

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Summary of Steps

1. Define the parabola and determine the slopes of relevant lines:
   • Tangent at $P(x_1, y_1)$.
   • Line joining $P$ to the focus.
   • Line parallel to the axis through $P$.
2. Compute the angles using the tangent formula.
3. Show that the angles of incidence and reflection are equal:

$$\theta_1 = \theta_2$$