angle between tangents

Question

if two tangents are drawn from a point on x² + y² = 16 to the circle x² + y² = 8 then the angle between the tangents is

(I)90⁰

(ii)45⁰

(iii)120⁰

(iv)180⁰

Answer 1

The radius of the circle x^2 + y^2 = 8 to the point of tangency will ,of course, be perpendicular to the tangent line.   So you have a right angle triangle with a hypotenuse of 4 and a leg of 2√2.  

 

Using the Pythagorean Theorem  the length of the other side can be calculated.

4^2 = (2√2)^2 + b^2

16 = 8 + b^2

8 = b^2

2√2 = b

 

So it is not only a right triangle, but an isoceles triangle.  A right isoceles triangle has base angles of 45º.  Since the hypotenuse is the bisector of the angle between the tangents,

 

the angle between the tangents is 2(45º) = 90º