Given the circle #(x-4)^2 + (y+6)^2 =4#, determine all values of the real number c if the direction #3x+2y=c# is the tangent of the circle?

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Answer 1 · 2016-09-11T11:26:07

#c=+-sqrt13#.

Observe that the Centre #C# of the circle is

#C(4,-6)", and, radius r=2"#.

If the given line # L : 3x+2y-c=0# is a tgt. to the circle, then, from

Geometry, we know that,

#"The" bot"-distance from "C" to "L=r#.

Recall that

#"The" bot"-distance from "(x_1,y_1)" to line" : lx+my+n=0 "is"

# |lx_1+my_1+n|/sqrt(l^2+m^2)#.

Hence, # |3*4+2(-6)-c|/sqrt(9+4)=2, i.e., |c|=2sqrt13#.

#:. c=+-sqrt13#.