# Find the center of a circle given a tangent line & point

In summary, the conversation is about finding the center and radius of a circle that passes through A(1,1) and is tangent to the line y=2x-3 at the point B(3,3). The user attempted to use an equation from a website but got an incorrect answer. They also discussed the importance of including the point A(1,1) in the equation to accurately find the center of the circle.

## Homework Statement

"Find the center and radius of the circle that passes through A(1,1) and is tangent to the line y=2x-3 at the point B(3,3). (Picture of the graph: https://imgur.com/a/0wAnqcU)

## The Attempt at a Solution

Soo, I've been tryiing to use the equation shown in the link above, but i get an answer that i don't quite get. In the website where i found this (https://math.stackexchange.com/ques...ntre-of-circle-with-equation-of-tangent-given) it told me to use the coordinates where the tangent touches the circle.

I used the points (3,3) and substitute it with the x0 and the y0, and substitute the m and c with 2 and 3 respectively. When i simplify the equation, i get 15/3 for both x and y points, which will result to another (3,3). I might've missed something but the resulted coordinates isn't the center of the given circle.

And I don't think that's the formula you need: That formula only consists of the point where the circle touches the tangent and the linear equation. What about A(1,1)? Although I can't fully understand the formula, but just knowing the tangent point and equation won't be enough to make a circle(you can actually but you will make a infinite amount of them since A(1,1) doesn't matters), you need to find a formula that somehow takes the A(1,1) into a count.

Last edited:
You can get equations for 2 diameters out of the given information. Then solve simultaneously for the centre.

YoungPhysicist

## 1. How do you find the center of a circle given a tangent line and a point?

To find the center of a circle, you will need to first draw a line perpendicular to the tangent line at the given point. Next, you will need to find the midpoint of this perpendicular line, which will be the center of the circle.

## 2. What information do you need to find the center of a circle?

You will need the tangent line and one point that lies on the circle in order to find the center.

## 3. Can you find the center of a circle using only a tangent line?

No, you will need at least one point on the circle in order to find the center.

## 4. What is the formula for finding the center of a circle given a tangent line and a point?

The formula for finding the center of a circle is to draw a line perpendicular to the tangent line at the given point, and then find the midpoint of this perpendicular line. This midpoint will be the center of the circle.

## 5. Can you use any point on the tangent line to find the center of a circle?

No, you will need a point that lies on the circle in order to find the center. A point on the tangent line alone will not be enough information to find the center.

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