07_Ex 6.7

Exercise Questions

Questions	Links
Q1. Find equation of the tangent and normal to…	6.7 Q-1
Q2. Write equation of the tangent to…	6.7 Q-2
Q3. Find equation of tangent which passes…	6.7 Q-3
Q4. Find equations of the normal to the parabola…	6.7 Q-4
Q5. Find equation of tangent to the ellipse…	6.7 Q-5
Q6. Find equation of tangent to the conic…	6.7 Q-6
Q7. Find equation of common tangents to the…	6.7 Q-7
Q8. Find the points of intersection of the given conics…	6.7 Q-8

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Overview

This exercise focuses on finding the equations of tangents and normals to various conic sections at specific points. It also covers the application of differentiation to determine slopes of tangents and normals, as well as solving geometric problems related to conics such as ellipses, parabolas, and hyperbolas. These concepts are fundamental in calculus, specifically in the study of curves and their properties.

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Key Concepts

1. Tangents and Normals to Conics: The tangent to a conic at a point is a straight line that touches the curve at that point without crossing it. The normal is a line perpendicular to the tangent at the point.

2. Differentiation of Conic Equations: To find the slope of the tangent or normal, we differentiate the equation of the conic with respect to $x$. The slope of the tangent at a point is given by $\frac{dy}{dx}$, while the normal slope is the negative reciprocal of the tangent’s slope.

3. Standard Forms of Conics:

   • Parabola: $y^2 = 4ax$
   • Ellipse: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$
   • Hyperbola: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$
   • Circle: $x^2 + y^2 = r^2$

4. Equation of Tangent and Normal: The general form for the tangent line at a point $(x_1, y_1)$ on a conic is given by:

$$y - y_1 = m(x - x_1)$$

where $m$ is the slope of the tangent.

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Important Formulas

1. Tangent Slope for Parabola $y^2 = 4ax$:

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

2. Tangent Slope for Ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$:

$$\frac{dy}{dx} = -\frac{b^2 x}{a^2 y}$$

3. Tangent Slope for Hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$:

$$\frac{dy}{dx} = \frac{b^2 x}{a^2 y}$$

4. Normal Slope: The slope of the normal is the negative reciprocal of the tangent slope:

$$m_{\text{normal}} = -\frac{1}{m_{\text{tangent}}}$$

5. Equation of Normal: The normal line at a point $(x_1, y_1)$ on a conic is given by:

$$y - y_1 = m_{\text{normal}}(x - x_1)$$

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Tips and Tricks

• For a parabola, remember that the slope of the tangent at any point is inversely proportional to the y-coordinate of the point.
• For ellipses and hyperbolas, the slope formula involves both $x$ and $y$ coordinates, making it crucial to differentiate properly.
• To simplify finding the equation of the tangent or normal, always substitute the coordinates of the given point into the derivative first to find the slope, then use the point-slope form to derive the line equation.
• When dealing with problems involving common tangents, set the discriminant of the quadratic equation formed equal to zero to ensure that the line touches both curves at exactly one point.

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Summary

In this exercise, we learned how to compute the equations of tangents and normals to different conics. By applying differentiation, we determined the slopes of these lines at given points on the curves. We also explored how to handle conics in standard forms such as parabolas, ellipses, and hyperbolas, and how to use these principles to solve problems involving geometric properties like parallelism and intersections of tangents.

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Reference

By Sir Shahzad Sair: