Lingkaran Bagian 6 - Menentukan Persamaan Garis Singgung Lingkaran Jika Diketahui Gradiennya

Introduction

This tutorial will guide you through determining the equation of a tangent line to a circle when the gradient of the tangent line is known. This is particularly useful for high school mathematics students studying conic sections and tangents in circles.

Step 1: Understand the Basics of Tangents and Circles

• A tangent line to a circle touches the circle at exactly one point.
• The gradient (slope) of the tangent line is perpendicular to the radius at the point of tangency.
• The general equation of a circle can be represented as: [ (x - h)^2 + (y - k)^2 = r^2 ] where ((h, k)) is the center of the circle and (r) is the radius.

Step 2: Identify the Given Information

• You need to know:
  • The coordinates of the center of the circle ((h, k)).
  • The radius (r).
  • The gradient (m) of the tangent line.

Step 3: Calculate the Gradient of the Radius

• The gradient of the radius from the center to the point of tangency is given by: [ m_{radius} = -\frac{1}{m} ] where (m) is the gradient of the tangent.

Step 4: Determine the Point of Tangency

• Use the formula for the distance between two points to find the point of tangency ((x_1, y_1)): [ \sqrt{(x_1 - h)^2 + (y_1 - k)^2} = r ] By substituting the coordinates into the equation, you can express (y_1) in terms of (x_1) using the tangent gradient.

Step 5: Formulate the Equation of the Tangent Line

• The equation of the tangent line can be expressed in point-slope form: [ y - y_1 = m(x - x_1) ] Simplifying this will give you the final equation of the tangent line.

Step 6: Verify Your Solution

• Ensure that the point of tangency lies on both the circle and the tangent line.
• Substitute the coordinates of the point of tangency back into the circle's equation to confirm it satisfies the equation.

Conclusion

In this tutorial, you learned how to determine the equation of a tangent line to a circle given its gradient. Key steps include understanding the relationship between the tangent and radius gradients, identifying the point of tangency, and formulating the tangent line equation. For further practice, consider working through example problems or reviewing related videos on circle properties and equations.