MATH (CIRCLE PERPERDICULAR) PERPENDICULAR

Unit Structure

• The unit is titled "applications" and consists of several topics:

  • Straight Lines

  • Circles

  • Integration

  • Recurrence Relations

Circle Definitions and Concepts

• A circle can be defined using two pieces of information:

  • Center (c): This is a point defined by coordinates (a, b).

  • Radius (r): This is the distance from the center to any point on the circle.

Equation of a Circle

• The general equation of a circle can be given as:

  • If center = (a, b) and radius = r, then the equation of the circle is:
    $(x - a)^2 + (y - b)^2 = r^2$

Example 1

• Given:

  • Center at (3, 2)

  • Radius = 5

• Equation derived: $(x - 3)^2 + (y - 2)^2 = 5^2$

  • Simplified to:
    $(x - 3)^2 + (y - 2)^2 = 25$

Example 2

• Given:

  • Center at (-5, -2)

  • Radius = 10

• Equation derived: $(x - (-5))^2 + (y - (-2))^2 = 10^2$

  • Simplified to:
    $(x + 5)^2 + (y + 2)^2 = 100$

Checking Points Against Circles

• To determine if a point lies on a circle:

  • Substitute the point's coordinates (x, y) into the circle's equation.

  • If the left-hand side equals the right-hand side, then the point lies on the circle; otherwise, it does not.

Example 3

• Assess if the point (1, 1) lies on the circle defined by center (3, -6) and radius = 6:

  • Substitute:
    $(1 - 3)^2 + (1 + 6)^2 = 6^2$

  • Calculation:

  • Left Side:
    $(-2)^2 + (7)^2 = 4 + 49 = 53$

  • Right Side:
    $36$

  • Conclusion: Point (1, 1) does not lie on the circle, as 53 ≠ 36.

Straight Lines and Their Equations

• The standard form of the equation for straight lines is:

  • $y - b = m(x - a)$ where:

  • m = slope (gradient)

  • (a, b) = a point through which the line passes

Finding Straight Line Equations

• Gradient (m) indicates the steepness of a line and is calculated based on coordinates.

• If given a straight line's equation in slope-intercept form, like $y = mx + c$, where m is the gradient and c is the y-intercept:

  • Example: For $y = 4x + 2$, the gradient = 4.

Parallel Lines

• Two lines are parallel if they have the same gradient.

• Mathematical interpretation:

  • If lines L1 and L2 both have the gradient m, they will never intersect.

  • For example, both having a gradient of 2 indicates they are parallel.

Example of Parallel Line Calculation

• Given: Find the equation of the line parallel to $y = 2x + 1$ and passing through (1, 3).

  • Since it is parallel, it will have the same gradient (m = 2):

  • Using the line equation:
    $y - 3 = 2(x - 1)$
    Solution leads to:
    $y = 2x + 1$

Perpendicular Lines

• Perpendicular lines intersect at 90 degrees.

• The relationship between gradients (m1 and m2) of two perpendicular lines is given by:

  • $m<em>1 imes m</em>2 = -1$

Finding Perpendicular Gradients

• If the gradient of a line is $m$, then the gradient of the line perpendicular to it is:

  • $- rac{1}{m}$

Example 4

• Given line: $y = rac{2}{3}x + 4$, gradient = $rac{2}{3}$.

• Perpendicular gradient = $- rac{1}{ rac{2}{3}} = - rac{3}{2}$.

Example of Perpendicular Line Calculation

• Given: Find the equation of the line perpendicular to $y = rac{1}{2}x + 1$ and passing through (2, 3).

  • Gradient = $-2$ (obtained from flipping $( rac{1}{2})$ and changing the sign).

  • Apply the point-slope formula:
    $y - 3 = -2(x - 2)$

  • Leads to:
    $y = -2x + 7$

Graphical Representation of Relationships

• The content discusses visual properties, such as intersections and tangents:

  • Intersection of Line and Circle: Can occur at 0 (tangent), 1 (point of contact), or 2 (cutting through).

  • Tangents are lines that touch the circle at exactly one point and are conceptually different from sine, cosine, and tangent functions.

Practice Exercises

• Various problems based on the equations of circles, checking point inclusion, straight lines, parallel lines, and perpendicular lines are provided for drill work.

• These exercises help reinforce understanding of definitions, calculations, and the relationships between gradients, lines, and circles.