The Equation of a Diameter

Oct 13, 2025 by recoveryshake.com 89 views

Introduction

When a circle is plotted on a coordinate plane, the equation of the diameter that passes through a point on the circle can be used to determine the equation of a tangent line. In this article, we will explore how to find the equation of a tangent line that is tangent to a given circle.

The Equation of a Diameter

The equation of the diameter that passes through point Q on the circle is given as y = 4x + 2. This equation represents a line that passes through the center of the circle and is perpendicular to the radius that intersects point Q.

The Slope of the Diameter

To find the slope of the diameter, we can use the equation of the diameter, which is y = 4x + 2. The slope of this line is 4, which means that for every unit increase in x, the value of y increases by 4 units.

The Slope of the Tangent Line

Since the tangent line is perpendicular to the radius that intersects point Q, the slope of the tangent line is the negative reciprocal of the slope of the radius. In this case, the slope of the radius is 4, so the slope of the tangent line is -1/4.

The Equation of the Tangent Line

Using the point-slope form of a line, we can write the equation of the tangent line as y - y1 = m(x - x1), where m is the slope of the line and (x1, y1) is a point on the line. In this case, the slope of the tangent line is -1/4, and the point (x1, y1) is the point of tangency.

Finding the Point of Tangency

To find the point of tangency, we need to find the point where the tangent line intersects the circle. Since the tangent line is perpendicular to the radius that intersects point Q, the point of tangency will be on the line that passes through the center of the circle and is perpendicular to the radius.

Using the Equation of the Circle

The equation of the circle is x^2 + y^2 = r^2, where r is the radius of the circle. We can substitute the equation of the tangent line into this equation to find the point of tangency.

Substituting the Equation of the Tangent Line

Substituting the equation of the tangent line, y = -1/4x + b, into the equation of the circle, we get x^2 + (-1/4x + b)^2 = r^2.

Expanding the Equation

Expanding the equation, we get x^2 + 1/16x^2 - 1/2bx + b^2 = r^2.

Simplifying the Equation

Simplifying the equation, we get (1 + 1/16)x^2 - 1/2bx + (b^2 - r^2) = 0.

Solving for x

Solving for x, we get x = (-b ± √(b^2 - 4(1 + 1/16)(b^2 - r^2))) / (2(1 + 1/16)).

Finding the Value of b

To find the value of b, we need to find the point where the tangent line intersects the circle. Since the tangent line is perpendicular to the radius that intersects point Q, the point of tangency will be on the line that passes through the center of the circle and is perpendicular to the radius.

Using the Equation of the Diameter

The equation of the diameter is y = 4x + 2. We can substitute this equation into the equation of the tangent line to find the value of b.

Substituting the Equation of the Diameter

Substituting the equation of the diameter, y = 4x + 2, into the equation of the tangent line, y = -1/4x + b, we get 4x + 2 = -1/4x + b.

Solving for b

Solving for b, we get b = 4x + 2 + 1/4x.

Simplifying the Equation

Simplifying the equation, we get b = (16x + 8 + x) / 4.

Simplifying Further

Simplifying further, we get b = (17x + 8) / 4.

Substituting the Value of b

Substituting the value of b into the equation of the tangent line, we get y = -1/4x + (17x + 8) / 4.

Simplifying the Equation

Simplifying the equation, we get y = (-x + 17x + 8) / 4.

Simplifying Further

Simplifying further, we get y = (16x + 8) / 4.

Simplifying Even Further

Simplifying even further, we get y = 4x + 2.

Conclusion

In conclusion, the equation of a line that is tangent to circle P is y = 4x + 2. This equation represents a line that passes through the center of the circle and is perpendicular to the radius that intersects point Q.

Final Answer

The final answer is y = 4x + 2.

Introduction

In our previous article, we explored how to find the equation of a tangent line that is tangent to a given circle. In this article, we will answer some frequently asked questions related to the topic.

Q: What is the equation of a tangent line that is tangent to circle P?

A: The equation of a tangent line that is tangent to circle P is y = 4x + 2.

Q: How do I find the equation of a tangent line that is tangent to a given circle?

A: To find the equation of a tangent line that is tangent to a given circle, you need to find the point of tangency and then use the point-slope form of a line to write the equation of the tangent line.

Q: What is the point of tangency?

A: The point of tangency is the point where the tangent line intersects the circle.

Q: How do I find the point of tangency?

A: To find the point of tangency, you need to find the point where the tangent line intersects the circle. This can be done by substituting the equation of the tangent line into the equation of the circle.

Q: What is the equation of the circle?

A: The equation of the circle is x^2 + y^2 = r^2, where r is the radius of the circle.

Q: How do I substitute the equation of the tangent line into the equation of the circle?

A: To substitute the equation of the tangent line into the equation of the circle, you need to replace y with the equation of the tangent line in the equation of the circle.

Q: What is the equation of the tangent line in terms of x?

A: The equation of the tangent line in terms of x is y = -1/4x + b.

Q: How do I find the value of b?

A: To find the value of b, you need to find the point where the tangent line intersects the circle. This can be done by substituting the equation of the diameter into the equation of the tangent line.

Q: What is the equation of the diameter?

A: The equation of the diameter is y = 4x + 2.

Q: How do I substitute the equation of the diameter into the equation of the tangent line?

A: To substitute the equation of the diameter into the equation of the tangent line, you need to replace y with the equation of the diameter in the equation of the tangent line.

Q: What is the final equation of the tangent line?

A: The final equation of the tangent line is y = 4x + 2.

Q: What is the significance of the equation of the tangent line?

A: The equation of the tangent line represents a line that passes through the center of the circle and is perpendicular to the radius that intersects point Q.

Q: What is the relationship between the equation of the tangent line and the equation of the circle?

A: The equation of the tangent line is perpendicular to the radius that intersects point Q, which means that the equation of the tangent line is perpendicular to the equation of the circle.

Q: How do I use the equation of the tangent line to find the equation of the circle?

A: To use the equation of the tangent line to find the equation of the circle, you need to substitute the equation of the tangent line into the equation of the circle and solve for the radius of the circle.

Q: What is the final answer?

A: The final answer is y = 4x + 2.

Conclusion

In conclusion, the equation of a tangent line that is tangent to circle P is y = 4x + 2. This equation represents a line that passes through the center of the circle and is perpendicular to the radius that intersects point Q. We hope that this Q&A article has been helpful in answering your questions related to the topic.