Salah satu persamaan garis singgung lingkaran x^2+y^2-2x+6y-10=0 yang sejajar dengan garis 2x-...

Summary

This transcript explains the mathematical process of determining the center and radius of a circle from its equation, along with the steps to derive the equation of its tangent line. The general equation of a circle is discussed, and the method for transforming it into a standard form is demonstrated. The transcript also covers the formula for the equation of a tangent line, highlighting the concept of parallel lines having the same gradient. By using these principles, the equation of the tangent to the given circle is determined, resulting in the solution that corresponds to the choice in the question.

Takeaways

• 😀 The general equation of a circle is (x - a)^2 + (y - b)^2 = r^2, where (a, b) is the center and r is the radius.
• 😀 The equation of the tangent line to a circle is y - b = m(x - a) ± r√(m² + 1), where (a, b) is the center and m is the gradient of the tangent.
• 😀 To determine the center and radius of a given circle, you need to transform the provided equation into the general form.
• 😀 The transformation process involves completing the square for both x and y terms.
• 😀 For the x term, complete the square by adjusting the equation to match the form (x - 1)².
• 😀 For the y term, complete the square by adjusting the equation to match the form (y + 3)².
• 😀 After completing the square, the equation of the circle becomes (x - 1)² + (y + 3)² = 20, revealing the center at (1, -3) and radius √20.
• 😀 The gradient of the tangent line is 2 because the equation of the tangent line is parallel to the line 2x - y + 4 = 0.
• 😀 The gradient of a tangent line is equal to the gradient of the line it is parallel to.
• 😀 The tangent equation can be derived using the general tangent formula, resulting in the equation 2x - y = -5 as one of the tangent lines.

Q & A

What is the general equation of a circle?

The general equation of a circle is (x - a)² + (y - b)² = r², where (a, b) is the center of the circle and r is the radius.

What is the equation for the tangent line to a circle?

The equation of the tangent line to a circle is (y - b) = m(x - a) ± r√(m² + 1), where (a, b) is the center of the circle, m is the slope of the tangent line, and r is the radius of the circle.

What is the first step in solving the circle's equation in the problem?

The first step is to convert the given circle's equation into the general form (x - a)² + (y - b)² = r² to determine the center (a, b) and the radius r.

How is the equation of the circle transformed in this problem?

The equation is transformed by completing the square for both x and y terms. For x, (x - 2)² is formed by adding and subtracting 1, and for y, (y + 3)² is formed by adding and subtracting 9.

What is the center and radius of the circle in this problem?

The center of the circle is at (1, -3), and the radius is 2√5.

What is the significance of the tangent line being parallel to the line 2x - y + 4 = 0?

The tangent line is parallel to the given line, meaning it has the same slope. This allows us to use the slope of the given line (m = 2) for the equation of the tangent line.

How is the slope (m) of the tangent line determined?

The slope of the given line is determined by rearranging the equation into the form y = mx + c. In this case, the equation 2x + 4 = y gives the slope m = 2.

What is the equation of the tangent line derived in this problem?

The equation of the tangent line is 2x - y = -5.

What formula is used to find the equation of the tangent line once the center, radius, and slope are known?

The formula used is (y - b) = m(x - a) ± r√(m² + 1), where m is the slope of the tangent, and a, b, r are the center and radius of the circle.

What is the importance of the steps where constants are added or subtracted in the process?

The steps of adding or subtracting constants are necessary to complete the square, which allows the equation of the circle to be rewritten in a form that easily reveals the center and radius.