- #1

Krushnaraj Pandya

Gold Member

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## Homework Statement

Equation of the circle passing through the point (1,2) and (3,4) and touching the line 3x+y-3=0 is?

## Homework Equations

x^2+y^2+2gx+2fy+c=0...(1)

(-g,-f)=center of circle

sqrt(g^2+f^2-c)=radius...(2)

## The Attempt at a Solution

Putting (1,2) and (3,4) in equation 1 we get

**5+2g+4f+c=0; 25+6g+8f+c=0.**

Now, line joining the two points will be perpendicular to the line joining center and midpoint of that line (chord perpendicular to radius). Say (h,k) is center, slope joining the two points is 1 so slope of radius through midpoint is -1 (perpendicular lines), midpoint of chord is (2,3); equating -1 to slope of (h,k) and (2,3) gives us k+h=5- but h= -g and k= -f; so

**-g-f=5**Solving these three equations gives c=40, f= -35/2 and g= 25/2 which is the wrong circle. I know there are other ways to solve this but I want to know why this method is not working in particular- I double checked all the calculations and I can't figure out anything wrong with my logic, Thank you for your help