# **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 Ellipses:** Calculating the points where a linear equation meets an elliptical curve and determining the resulting chord length. * **Condition of Tangency:** Applying the algebraic condition $c^2 = a^2m^2 + b^2$ for a line to be tangent to an ellipse. * **Tangent Equations:** Constructing tangent lines using both the point form $(x_1, y_1)$ and the slope form ($m$). * **Normal Equations:** Deriving the equation of the normal line perpendicular to the tangent at a specific point. * **General Second-Degree Ellipses:** Handling tangent and normal calculations for shifted or non-standard ellipse equations. * **Reflection Property:** Understanding the geometric relationship between tangents and the lines connecting the point of tangency to the foci. --- # **Important Formulas** Below are the key formulas used in this exercise: | Description | Formula | | :--- | :--- | | Standard Ellipse Equation | $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ | | Condition of Tangency ($y = mx + c$) | $c^2 = a^2m^2 + b^2$ | | Tangent at $(x_1, y_1)$ (Standard) | $\frac{xx_1}{a^2} + \frac{yy_1}{b^2} = 1$ | | Normal at $(x_1, y_1)$ (Standard) | $\frac{a^2x}{x_1} - \frac{b^2y}{y_1} = a^2 - b^2$ | | Tangent in Slope Form | $y = mx \pm \sqrt{a^2m^2 + b^2}$ | | General Tangent Substitution | $x^2 \to xx_1, \quad y^2 \to yy_1, \quad x \to \frac{x+x_1}{2}, \quad y \to \frac{y+y_1}{2}$ | --- # **Summary** This exercise covers the analytical geometry of the ellipse, specifically focusing on its linear interactions. Key strategies include using the **method of substitution** to find intersection points and chord lengths, applying the **tangency condition** to solve for unknown coefficients, and utilizing **point-form transformations** ($T=0$) to find tangents and normals for both standard and general second-degree equations. A significant takeaway is the **reflection property**, which states that the tangent at any point on an ellipse bisects the external angle between the focal radii.