Lines Tangent to a Circle Homework Study Notes

Lines Tangent to a Circle Homework Study Notes

Understanding Tangent Lines to Circles

• A tangent line to a circle is a line that touches the circle at exactly one point.

• The tangent line is perpendicular to the radius drawn to the point of tangency.

• The general form of a circle's equation is $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center and $r$ is the radius.

Problems Overview

• Throughout the problems, students are tasked with finding unknowns related to circles and tangent lines.

Problem 1

• Find $x$ to the nearest tenth using the following equation:
  $2(x) + 5 = 3(2 + x)$

• Steps to solve:
    1. Distribute: $2x + 5 = 6 + 3x$
    2. Combine like terms:
       $2x - 3x = 6 - 5$
       $-x = 1$
    3. Thus, $x = -1$ (to the nearest tenth is not applicable here).

Problem 2

• Perimeter of a polygon given side lengths. Example:
    - Sides are 2 in, 12 ft: P = 2(2) + 2(12) = 4 + 24 = 28 units.

Problem 3

• Find the perimeter of a triangle where the perimeter is given as 76.
  $P = 2(a) + 2(b) + 2(x)$
  Substituting values yields:
  $76 = 2(5) + 2(25) + 2(x)$
  Solving follows:
    - $76 = 10 + 50 + 2x$
    - $76 - 60 = 2x$
    - $16 = 2x <br>ightarrow x = 8$

Finding Tangent Lines

Problem 7

• Find the equation of a line tangent to the circle:
  $(x - 2)^2 + (y + 3)^2 = 25$ at the point $(—2, 0).$

1. Confirm the point lies on the circle by substituting into the equation.

2. Deriving the slope at the point requires implicit differentiation.

3. Write the equation of the tangent line using point-slope form: $y - y_1 = m(x - x_1)$.

Problem 8

• Find the equation tangent to the circle at given coordinates $(2, -7)$ in the equation:
  $(x + 3)^2 + (y - 5)^2 = 169$

1. Check if the point lies on the circle.

2. Differentiate to find slope, then use point-slope form for the tangent.

Problem 9

• Given the equation of the circle:
  $(x + 3)^2 + (y + 4)^2 = 17$
  and the line tangent:
  $y = 4x + 25$, find the point of tangency.

1. Set the equation of the line equal to that of the circle to find intersecting coordinates.

Problem 10

• Derive the tangent line for a circle centered at $a$ with radius 5 passing through some point $(x_1, y_1)$.

1. Coordinate for tangent point on circle must satisfy:
   $d = ext{radius}$.

Problem 11

• Find tangent line for the circle centered at $(1, 2)$ with radius 2, passing through the point $(0, 3)$.

1. Use the distance formula to find the vertical distance and derive tangent lines based on the coordinates.

Summary of Deriving Equations

• Tangent line equations are derived through:
    - Determining slopes with implicit differentiation.
    - Employing point-slope form for tangent lines based on derived points (tangent points) and slopes.

• Checking if coordinates lie on the respective circles is essential in solving for tangent equations.