The equation of the tangent line to the circle
x2+y2+2x−6y+2=0
that is parallel to the line x−y+3=0 is ....
Explanation
Given circle L=x2+y2+2x−6y+2=0 parallel to line x−y+3=0
Determining the Circle's Radius
The general form of a circle (x−a)2+(y−b)2=r2 can be found from equation L
x2+y2+2x−6y+2=0
(x+1)2−1+(y−3)2−9=−2
(x+1)2+(y−3)2=−2+10
(x+1)2+(y−3)2=8
Therefore, the circle's center is at (−1,3) and radius r=8=22
Determining the Gradient
Since y=mx+c and the line is parallel to the tangent line, the gradient is
x−y+3=0⇔y=x+3
mg=m=1
Tangent Line Equation
The formula for the tangent line equation to a circle with gradient m
y−b=m(x−a)±r1+m2
Substitute the known values
y−3=1(x+1)±8⋅1+(1)2
y−3=x+1±8⋅2
y−3=x+1±4
We obtain two equations
y=x+4+4ory=x+4−4
y=x+8ory=x
Convert to general form
x−y+8=0orx−y=0
The most appropriate answer is C: x−y+8=0