If (-2,-3) to the circle x2+y2+3=0 what's is tangent length?

1 Answer
Douglas K.
Apr 27, 2018

The length of the tangent is #sqrt10#

Explanation:

I shall assume that you mean #x^2+y^2=3# for the circle.

Comparing #x^2+y^2=3#

To the general Cartesian equation of a circle:

#(x-h)^2+(y-k)^2=r^2#

We observe that the center, #(h,k)#, of the circle is #(0,0)# and the radius, #r#, of the circle is #sqrt3#

The length of the line from #(-2,-3)# to the center of the circle #(0,0)#:

#c = sqrt((-2-0)^2+(-3-0)^2)#

#c = sqrt(4+9)#

#c = sqrt13#

The tangent line, #t#, the radius, #r#, and the line from the point to the center, #c#, form a right triangle, therefore, we can use the Pythagorean Theorem:

#c^2 = t^2+r^2#

#(sqrt13)^2 = t^2+(sqrt3)^2#

#13 = t^2+3#

#t^2 = 10#

#t = sqrt10#

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