Length of Tangent on a Circle

Length of Tangent on a Circle

The length of a tangent to a circle depends on two factors: the radius of the circle and the distance between the point of tangency and the centre of the circle.

If you have a circle with radius “r” and you draw a tangent line from a point outside the circle to the point of tangency, the length of the tangent can be calculated using the Pythagorean theorem.

Let’s assume that the distance between the centre of the circle and the point of tangency is “d”. The length of the tangent line, denoted as “t,” can be calculated as follows:

In this equation, d represents the distance from the center of the circle to the point of tangency, and r represents the radius of the circle.

It’s important to note that if the point of tangency is inside the circle, there won’t be any tangent lines.

What is the Length of the Tangent?

The length of a tangent is a geometric concept that depends on the specific context in which it is used. In general, a tangent is a line that intersects a curve or circle at a single point, and its length can vary depending on the size and shape of the curve or circle.

If you are referring to the tangent line to a circle, the length of the tangent can be calculated using the Pythagorean theorem. If a line is tangent to a circle at a point, and the radius of the circle is known, then the length of the tangent can be determined. In this case, the length of the tangent is equal to the square root of the product of the lengths of the two segments created by the point of tangency. One segment is the radius of the circle, and the other segment is the distance from the point of tangency to the point where the tangent intersects the circle.

It’s worth mentioning that the length of the tangent can also refer to the length of a straight line segment drawn from an external point to a circle, such that the line segment is tangent to the circle. Again, the length of this tangent line segment will depend on the specific parameters of the circle and the external point.

What is Tangent to a Circle?

In geometry, a tangent to a circle is a line that intersects the circle at exactly one point. This point of intersection is called the point of tangency. The tangent line is perpendicular to the radius of the circle at the point of tangency.

The tangent line has the special property that it touches the circle at just one point and does not pass through the circle. If a line intersects a circle at more than one point, it is not a tangent.

The tangent line to a circle can be thought of as the limiting case of a secant line as the two points where the secant intersects the circle approach each other. In this limiting case, the two points of intersection merge into a single point, and the secant becomes a tangent.

Tangents to a circle are often used in geometric constructions, calculations involving circles, and in solving problems related to circles and their properties. They have applications in various fields, including physics, engineering, and computer graphics.

Theorems Related to Length of Tangent

There are several theorems related to the length of tangents drawn to circles. Here are a few notable ones:

Theorem: Tangent segments drawn from an external point to a circle are congruent.

This theorem states that if you draw two tangents from an external point to a circle, the lengths of the tangent segments (the line segments from the point of tangency to the external point) are equal. In other words, the two tangents are congruent.

Theorem: The length of a tangent segment drawn from an external point to a circle is equal to the length of the radius perpendicular to that tangent.

This theorem relates the length of a tangent segment to the radius of the circle. If you draw a tangent from an external point to a circle, and then draw a radius from the center of the circle to the point of tangency, the length of the tangent segment is equal to the length of the radius perpendicular to that tangent.

Theorem: The lengths of tangents drawn from an external point to a circle are equal if and only if they are congruent.

This theorem establishes the equivalence between the equality of lengths and the congruence of tangents. If two tangents drawn from an external point to a circle have equal lengths, then they are congruent. Conversely, if two tangents are congruent, then their lengths are equal.

These theorems provide useful insights into the properties of tangents drawn to circles and their relationship with the circle’s radius. They are often applied in geometry problems and constructions involving circles.

Equations on Tangent on a Circle

The equations of tangents to a curve depend on the specific curve and the point at which the tangent is drawn. Here are the general equations for finding the equation of the tangent line to a curve at a given point:

For a curve defined by a function y = f(x):

a. Find the derivative of the function, f'(x), which represents the slope of the tangent line at any point on the curve.

b. Substitute the x-coordinate of the given point into f'(x) to find the slope of the tangent line at that point.

c. Use the point-slope form of a linear equation, y – y₁ = m(x – x₁), where (x₁, y₁) is the given point and m is the slope of the tangent line, to obtain the equation of the tangent line.

For a curve defined parametrically by x = g(t) and y = h(t):

a. Find the derivatives dx/dt and dy/dt, which represent the rates of change of x and y with respect to the parameter t.

b. Substitute the value of t corresponding to the given point into dx/dt and dy/dt to find the slopes of the tangent line at that point.

c. Use the point-slope form of a linear equation, y – y₁ = m(x – x₁), where (x₁, y₁) is the given point and m is the slope of the tangent line, to obtain the equation of the tangent line.

It’s important to note that these equations give the equations of the tangent lines at specific points on the curve. The actual equations will vary depending on the specific curve being considered.

Solved Examples on Tangent to a Circle

Here are some solved examples on finding tangents to a circle:

Example 1:

Find the equation of the tangent to the circle x^2 + y^2 = 25 at the point (3, 4).

Solution:

The equation of a circle with center (h, k) and radius r is given by (x – h)^2 + (y – k)^2 = r^2.

Comparing this equation to x^2 + y^2 = 25, we can see that the center of the circle is at (0, 0) and the radius is 5.

To find the equation of the tangent, we need the slope of the tangent line and a point on the line.

The slope of the tangent is perpendicular to the radius at the point of tangency. Therefore, we can find the slope of the radius passing through (3, 4).

The slope of the radius is given by (y2 – y1) / (x2 – x1), where (x1, y1) is the center of the circle and (x2, y2) is the point on the radius.

Using (x1, y1) = (0, 0) and (x2, y2) = (3, 4), we get:

Slope = (4 – 0) / (3 – 0) = 4/3.

Since the slope of the radius is 4/3, the slope of the tangent will be the negative reciprocal, which is -3/4.

Now, we have the slope of the tangent (-3/4) and a point on the tangent (3, 4).

Using the point-slope form of the equation of a line, which is y – y1 = m(x – x1), we can substitute the values:

y – 4 = (-3/4)(x – 3).

Simplifying, we get:

y – 4 = (-3/4)x + (9/4).

Bringing everything to one side, we obtain:

3x + 4y – 25 = 0.

Therefore, the equation of the tangent to the circle x^2 + y^2 = 25 at the point (3, 4) is 3x + 4y – 25 = 0.

Example 2:

Find the equation of the tangent to the circle (x – 2)^2 + (y + 1)^2 = 9 at the point (4, -2).

Solution:

Comparing the equation to the standard form, we can see that the center of the circle is at (2, -1) and the radius is 3.

To find the equation of the tangent, we need the slope of the tangent line and a point on the line.

The slope of the tangent is perpendicular to the radius at the point of tangency. Therefore, we can find the slope of the radius passing through (4, -2).

The slope of the radius is given by (y2 – y1) / (x2 – x1), where (x1, y1) is the center of the circle and (x2, y2) is the point on the radius.

Using (x1, y1) = (2, -1) and (x2, y2) = (4, -2), we get:

Slope = (-2 – (-1)) / (4 – 2) = -1/2.

Since the slope of the radius is -1/2, the slope of the tangent will be the negative reciprocal, which is 2.

Now, we have the slope of the tangent (2) and a point on the tangent (4, -2).

Using the point-slope form of the equation of a line, we can substitute the values:

y – (-2) = 2(x – 4).

Simplifying, we get:

y + 2 = 2x – 8.

Bringing everything to one side, we obtain:

2x – y – 10 = 0.

Therefore, the equation of the tangent to the circle (x – 2)^2 + (y + 1)^2 = 9 at the point (4, -2) is 2x – y – 10 = 0.

These examples demonstrate how to find the equation of a tangent to a circle given the coordinates of the center, the radius, and the point of tangency.