The equation of the circle with center (a,b) and radius r is
(x−a)2+(y−b)2=r2 Differentiate both sides with respect to x
2(x−a)+2(y−b)y′=0 Solve for y′
y′=−y−bx−a The slope of the line tangent to the circle at (3,−3)
slope=m=−−3−b3−a=3+b3−a The equation of the tangent line tangent to the circle at (3,−3) is
y−(−3)=3+b3−a(x−3)
y=3+b3−ax−3⋅3+b3−a−3
y=3+b3−ax−3⋅3+b6+b−a
The line x−4y−15=0
y=41x−415 Then
⎩⎨⎧3+b3−a=41−3⋅3+b6+b−a=−415
⎩⎨⎧b=9−4a3−a6+b−a=5
b=9−4a,a∈R
Substitute
(x−a)2+(y−9+4a)2=r2
Point (3,−3)
(3−a)2+(−3−9+4a)2=r2
9−6a+a2+144−96a+16a2=r2
r2=17a2−102a+153
Point (6,2)
(6−a)2+(2−9+4a)2=r2
36−12a+a2+49−56a+16a2=r2
r2=17a2−68a+85
r2=17a2−102a+153=17a2−68a+85
34a=68
a=2
b=9−4(2)=1
r2=17(2)2−68(2)+85=17
The equation of the circle is
(x−2)2+(y−1)2=17
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