02_Ex 6.2

Exercise Questions

Questions	Links
Q1. Write down the equation of tangent and normal…	6.2 Q-1
Q2. Write down the equation of tangent and normal…	6.2 Q-2
Q3. Check the position of the point $(5,6)$…	6.2 Q-3
Q4. Find the length of tangent drawn…	6.2 Q-4
Q5. Find the length of the chord cut off…	6.2 Q-5
Q6. Find the coordinates of the point of intersection	6.2 Q-6
Q7. Find the equation of the tangent to the…	6.2 Q-7
Q8. Find equation of the tangents…	6.2 Q-8
Q9. Find an equation of chord of contact…	6.2 Q-9

---

Overview

In this exercise, we explore key concepts related to tangents and normals to circles, the position of points relative to circles, and the length of tangents and chords. These concepts are essential in understanding geometry, particularly when working with curves and lines in two-dimensional space. The formulas and techniques covered will aid in solving geometric problems involving circles and their tangents, normal lines, and chords.

---

Key Concepts

1. Circle Equation: The standard form of a circle’s equation is $x^2 + y^2 = r^2$, where $r$ is the radius of the circle.

2. Tangent to a Circle: A tangent to a circle at a point $(x_1, y_1)$ is a straight line that touches the circle at exactly one point. The slope of the tangent is given by $m_1 = -\frac{x_1}{y_1}$.

3. Normal to a Circle: The normal line to the circle at any point is perpendicular to the tangent. If the slope of the tangent is $m_1$, the slope of the normal is $m_2 = -\frac{1}{m_1}$.

4. Length of Tangent: The length of the tangent drawn from a point $(x_1, y_1)$ to a circle with equation $x^2 + y^2 + 2gx + 2fy + c = 0$ is calculated using the formula:

$$L = \sqrt{x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c}$$

5. Position of a Point Relative to a Circle: The position of a point $(x_0, y_0)$ relative to a circle is determined by substituting the point into the circle’s equation. The value $D = x_0^2 + y_0^2 - r^2$ helps to classify the position:

   • $D < 0$: The point is inside the circle.
   • $D = 0$: The point is on the circle.
   • $D > 0$: The point is outside the circle.

6. Length of a Chord: The length of the chord cut off from a line by a circle is given by:

$$L = 2 \sqrt{r^2 - d^2}$$

where $d$ is the perpendicular distance from the center of the circle to the line.

---

Important Formulas

1. Equation of a Circle:

$$x^2 + y^2 = r^2$$

where $r$ is the radius.

2. Slope of Tangent at $(x_1, y_1)$ on the circle $x^2 + y^2 = r^2$:

$$m_1 = -\frac{x_1}{y_1}$$

3. Slope of Normal to the circle at $(x_1, y_1)$:

$$m_2 = -\frac{1}{m_1}$$

4. Length of Tangent from a point $(x_1, y_1)$ to a circle:

$$L = \sqrt{x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c}$$

5. Position of a Point Relative to a Circle:
   Calculate $D = x_0^2 + y_0^2 - r^2$ to classify the point’s position:

   • $D < 0$: Inside
   • $D = 0$: On the circle
   • $D > 0$: Outside

6. Length of Chord from a line $Ax + By + C = 0$ to a circle $x^2 + y^2 = r^2$:

$$L = 2 \sqrt{r^2 - d^2}$$

where $d$ is the perpendicular distance from the center to the line.

---

Tips and Tricks

• Differentiation is key in finding the slope of the tangent to a circle. Remember, the slope is the derivative of the circle’s equation.

• Perpendicular lines: If you know the slope of a tangent, the normal’s slope will always be the negative reciprocal.

• Chord length: If you know the perpendicular distance from the center to the line, you can calculate the length of the chord easily using the formula $L = 2 \sqrt{r^2 - d^2}$.

• Using Symmetry: The equations of tangents often have a symmetrical property, especially when dealing with circles centered at the origin.

---

Summary

This exercise focuses on fundamental geometric concepts relating to circles, including the equation of the circle, tangents, normals, and the length of tangents and chords. Key formulas and techniques, such as using derivatives to find the slope of tangents and understanding the position of points relative to a circle, are central to solving problems in this area.

---

Reference

By Sir Shahzad Sair: