Lingkaran Bagian 5 - Persamaan Garis Singgung Melalui Titik Pada Lingkaran

Summary

In this educational video, Linda Yani teaches how to find the equation of a tangent line to a circle in various scenarios. She explains three common cases: tangents through a point on the circle, tangents when the gradient is known, and tangents through a point outside the circle. The video covers the necessary formulas for each case, including those for circles centered at the origin, in standard form, and in general form. Using clear examples, the video demonstrates step-by-step how to apply these formulas to find the equation of a tangent line. This makes it an essential guide for students learning about circle geometry.

Takeaways

• 😀 The video explains how to find the equation of a tangent line to a circle in three different cases: tangent through a point on the circle, tangent with a given slope, and tangent through a point outside the circle.
• 😀 The first case discusses how to find the equation of the tangent when the point lies on the circle using the formula: x₁x + y₁y = r².
• 😀 For a circle centered at (0,0), the equation is x² + y² = r², and the tangent equation through a point (x₁, y₁) on the circle is x₁x + y₁y = r².
• 😀 When the circle has a center at (a,b), the equation becomes (x - a)² + (y - b)² = r², and the tangent equation through a point (x₁, y₁) is (x₁ - a)(x - a) + (y₁ - b)(y - b) = r².
• 😀 For the second case, where the slope of the tangent is given, the formula involves the gradient of the tangent and the circle's equation to determine the tangent line.
• 😀 The third case focuses on finding the tangent through a point outside the circle. You must check whether the point lies inside or outside the circle before proceeding.
• 😀 The script emphasizes the importance of verifying that the point lies on the circle before using the formulas to find the tangent.
• 😀 Example 1 demonstrates finding the tangent to the circle x² + y² = 13 through the point P(2,3), with the solution: 2x + 3y - 13 = 0.
• 😀 Example 2 shows how to find the tangent to the circle (x - 2)² + (y + 4)² = 41 through the point P(-2, 1), resulting in the equation: -4x + 5y - 13 = 0.
• 😀 In Example 3, the equation of the tangent to the circle x² + y² + 12x - 6y + 13 = 0 through the point P(-2, -1) is found to be: -2x - y + 1 = 0.
• 😀 Linda encourages viewers to practice these formulas and problem-solving steps with additional examples to gain confidence in finding tangent equations.

Q & A

What are the three main types of tangent problems discussed in the video?

The three main types of tangent problems are: 1) Tangent through a point on the circle, 2) Tangent with a given slope, and 3) Tangent through a point outside the circle.

How do you find the equation of a tangent through a point on the circle?

To find the equation of a tangent through a point on the circle, you use the formula: x₁x + y₁y = r², where (x₁, y₁) is the point on the circle and r is the radius of the circle.

What is the formula for the equation of a tangent for a circle centered at (0, 0)?

For a circle centered at (0, 0) with equation x² + y² = r², the equation of the tangent through a point (x₁, y₁) on the circle is: x₁x + y₁y = r².

What formula do you use when the circle is in the standard form (x - a)² + (y - b)² = r²?

For a circle in the form (x - a)² + (y - b)² = r², the tangent formula is: (x₁ - a)(x) + (y₁ - b)(y) = r², where (x₁, y₁) is the point on the circle.

How do you find the equation of the tangent when the circle is in the general form x² + y² + ax + by + c = 0?

For a circle in the general form x² + y² + ax + by + c = 0, the equation of the tangent is: x₁x + y₁y + (a/2)x₁ + (b/2)y₁ + c = 0.

What is the first step in solving a tangent problem for a point on the circle?

The first step is to substitute the coordinates of the point into the equation of the circle to confirm that the point lies on the circle.

In the example where the circle is x² + y² = 13 and the point is P(2, 3), how do you confirm that the point lies on the circle?

Substitute the coordinates P(2, 3) into the equation x² + y² = 13. If the result equals 13, the point lies on the circle. In this case, 2² + 3² = 13, so the point is on the circle.

How do you find the tangent equation for the circle (x - 2)² + (y + 4)² = 41 through the point P(-2, 1)?

After confirming that the point P(-2, 1) lies on the circle, substitute into the tangent formula: (-4x + 5y - 13 = 0) to get the equation of the tangent.

What method can you use if you forget the formula for the tangent in the general form of the circle?

If you forget the tangent formula, you can convert the general form of the circle into its standard form by finding the center and radius, and then use the formula for tangents in standard form.

What are the benefits of converting the general form of the circle to the standard form before finding the tangent?

Converting the general form to the standard form can make it easier to identify the center and radius of the circle, which simplifies the process of finding the tangent equation.