Equation of Circle Passing Through Given Point and Line with Given Radius

[alijan kk]

Homework Statement

Find an equation of the circle passing through:

A(-3,1) with radius 2 and centre on the line 2x-3y+3=0

Homework Equations

x²+y²+2gx+2fy+c=0
r²=g²+f²-c

The Attempt at a Solution

using this equation , i have found 2 equations
-6g+2f+c=-10 by putting (-3,1)

-2g+3f+3=0 (-g,-f) in the linear equation which passes through the centre

how can i use radius information here to get the third equatio n ?

 

[mfb]

If A(-3,1) is on the circle, what do you know about its distance to the center?
Alternatively: Just plug in the radius in your second "relevant equation".

(It would be easier to start with the line of the centers and just one unknown variable, but your approach works as well).

 

[Mark44]

and centre at 2x-3y+3=0

I think you meant "and centre on (the line) 2x - 3y + 3 = 0."

 

[alijan kk]

If A(-3,1) is on the circle, what do you know about its distance to the center?
Alternatively: Just plug in the radius in your second "relevant equation".

(It would be easier to start with the line of the centers and just one unknown variable, but your approach works as well).

(-3,1) (-g,-f) distance between them is equal to r

g²+f²+2f+6g+10=r²

g²+f²-c=r²

i get a weird anwer by solving simultaneously ,,, like g=0

 

[alijan kk]

If A(-3,1) is on the circle, what do you know about its distance to the center?
Alternatively: Just plug in the radius in your second "relevant equation".

(It would be easier to start with the line of the centers and just one unknown variable, but your approach works as well).

(-3,1) (-g,-f) distance between them is equal to r

g²+f²+2f+6g+10=r²

g²+f²-c=r²

i get a weird anwer by solving simultaneously ,,, like g=0

which is the better and easier strategy ?

 

[mfb]

I would start with the suggestion at the end of my post. The center is on the line 2x-3y+3=0? Let its x value be x₀, calculate its y-value, calculate the distance to A(-3,1) and require that this distance is 2. A single variable, a single equation to satisfy.

 

[LCKurtz]

@alijan kk: Are you going to share with us what you got? You should have two answers.