Equation of Circle Centered at (-3,4) Touches Y-Axis

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Discussion Overview

The discussion revolves around finding the equation of a circle centered at (-3,4) that touches the y-axis. Participants explore various approaches to derive the equation, including geometric reasoning and algebraic methods.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • Some participants express confusion about how to start solving the problem, indicating a lack of instructions from a teacher.
  • One participant suggests using the geometric relationship between the center of the circle and the tangent point on the y-axis to find the radius.
  • Another participant provides the general formula for a circle and encourages sketching the situation to aid understanding.
  • A different approach involves setting up a system of equations to ensure the quadratic formed has one real root, leading to a specific value for the radius.
  • Several participants repeatedly ask how to begin, indicating a shared uncertainty about the problem-solving process.

Areas of Agreement / Disagreement

There is no consensus on a single method to solve the problem, and multiple approaches are presented. Participants express varying levels of understanding and confidence in their methods.

Contextual Notes

Some participants mention the need for a visual representation to clarify the problem, while others focus on algebraic manipulation. The discussion reflects a range of assumptions about prior knowledge and problem-solving strategies.

Who May Find This Useful

This discussion may be useful for students learning about the equations of circles, particularly those who are struggling with geometric interpretations and algebraic formulations.

thorpelizts
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Find the equation of the circle whose centre is (-3,4) and which touches the y-axis.

how do i even begin?
 
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thorpelizts said:
Find the equation of the circle whose centre is (-3,4) and which touches the y-axis.
how do i even begin?
Teacher gave no instructions, no teaching?
Google "equation of circle".
 
thorpelizts said:
Find the equation of the circle whose centre is (-3,4) and which touches the y-axis.

how do i even begin?

Hi thorpelizts, :)

Let \(P\equiv (-3,4)\) and let \(Q\) be the point of intersection of the circle and the y-axis. Since the y-axis is a tangent to the circle, \(PQ\) is perpendicular to the y-axis. Now I am sure you can find the length of \(PQ\) which is the radius of the circle. Can you give it a try?

Kind Regards,
Sudharaka.
 
Hello, thorpelizts!

Find the equation of the circle whose centre is (-3,4) and which touches the y-axis.

How do i even begin? . Make a sketch!
You are expected to know this formula: .(x-h)^2 + (y-k)^2 \:=\:r^2
. . where (h,k) is the center and r is the radius.

Code: 
                            |
                * * *       |
            *           *   |
          *               * |
         *                 *|
                            |
        *              r    *
        *         * - - - - *4
        *      (-3,4)       *
                            |
         *                 *|
          *               * |
            *           *   |
                * * *       |
                            |
    - - - - - - - + - - - - + - - -
                 -3         |
You know h = -3,\;k=4.

Can you guess what the radius is?
 
While soroban's method is easiest, you might also consider we want the solution of the systerm:

(x + 3)2 + (y - 4)2 = r2

x = 0

to have one real root.

Substitute into the first equation from the second:

(0 + 3)2 + (y - 4)2 = r2

(y - 4)2 + 9 - r2 = 0

We want this quadratic to have one root, hence the discriminant must be zero:

02 - 4(1)(9 - r2) = 0

r = 3
 
thorpelizts said:
Find the equation of the circle whose centre is (-3,4) and which touches the y-axis.

how do i even begin?

You begin by drawing a picture.

CB
 

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