# **Exercise Questions** All questions in this exercise are listed below. Click on a question to view its solution. --- --- # **Key Concepts** This exercise focuses on the following concepts: * **Intersection of Lines and Parabolas:** Solving simultaneous linear and quadratic equations to find points of contact or intersection. * **Condition of Tangency:** Determining the mathematical criteria for a line to touch a parabola at exactly one point. * **Tangent and Normal Equations:** Deriving and applying formulas for lines tangent and normal to a parabola at specific points. * **Geometric Properties:** Exploring relationships between tangents, focal chords, the directrix, and the latus rectum. * **Harmonic Mean in Parabolas:** Understanding the relationship between segments of a focal chord and the semi latus rectum. --- # **Important Formulas** Below are the key formulas used in this exercise: | Description | Formula | | :--- | :--- | | Tangent to $y^2 = 4ax$ at $(x_1, y_1)$ | $yy_1 = 2a(x + x_1)$ | | Normal to $y^2 = 4ax$ at $(x_1, y_1)$ | $y - y_1 = -\frac{y_1}{2a}(x - x_1)$ | | Condition of tangency ($y = mx + c$) | $c = \frac{a}{m}$ | | Point of contact for $y = mx + c$ | $(\frac{a}{m^2}, \frac{2a}{m})$ | | Harmonic mean of focal segments | $\frac{1}{SP} + \frac{1}{SQ} = \frac{1}{a}$ | --- # **Summary** This exercise covers the coordinate geometry of the parabola, specifically focusing on its interaction with straight lines. Key learnings include the method of finding intersection points, applying the condition of tangency to find unknown parameters, and constructing equations for tangents and normals. A significant portion of the exercise is dedicated to proving geometric theorems, such as the harmonic mean property of focal chords and angular properties of tangents, which are fundamental to understanding the reflective and structural nature of parabolic curves.