Find the equation of the circle passing through the point (2,3) and touching the line 2x+3y=4 at the point (2,0) - Maths -

Question

Find the equation of the circle passing through the point (2,3) and
touching the line 2x+3y=4 at the point (2,0)

Vijay Kumar Gupta · Accepted Answer

Dear Student,

Let the equation of circle be,      x-h2+y-k2=r2     
1where h, k=centre and r=radiusSince the circle passes through the points 2, 3Put x=2, y=3 in 1      2-h2+3-k2=r2     
2Also the circle passes through the points 2, 0Therefore,      2-h2+0-k2=r2     
3Substract 2 from 3, we get      k2-3-k2=0      k2-9+k2-6k=0      k2-9-k2+6k=0      6k=9      k=32Put this value in equation 3      2-h2+322=r2      2-h2+94=r2    
4Therefore from 1      x-h2+y-322=r2    Differentiate both side with respect to x      2x-h+2y-32dydx=0  At 2, 0 we have      22-h+20-32dydx=0        22-h-32dydx=0       dydx=2h-23Also from equation of tangent 2x+3y=4Slope of tangent=-23Therefore, -23=2h-23  h-2=-1  h=1Put the value of h in equation 4      2-12+94=r2       1+94=r2       r2=134Thus, equation of circle is      x-12+y-322=134 
Regards

Find the equation of the circle passing through the point (2,3) and touching the line 2x+3y=4 at the point (2,0).