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Circles

Circles and tangents

Question
If from any point on the circle x2 + y2 +2gx + 2fy + c = 0, tangents are drawn to the circle x2 + y2 + 2gx + 2fy + csin2α + (g2 + f2)cos2 α = 0, then angle between the tangents is 

Concepts used
- Circle with equation x2 + y2 + 2gx + 2fy + c = 0, has center at (-g,-f) and radius √(g2+f2-c),
√ (g2+f2-c) 
- sin2 + cos2 = 1

Answer
Rearrange the equation of circles to find the centers and radii
Circle 1 - x2 + y2 +2gx + 2fy + c = 0
x2 + y2 +2gx + 2fy + c = 0
x2 + y2 +2gx + 2fy + g2 + f2 = g2 + f2 - c 
(x+g)2 + (x+f)2 = (√(g2+f2-c))2

Circle 2 - x2 + y2 + 2gx + 2fy + csin2α + (g2 + f2)cos2 α = 0
x2 + y2 + 2gx + 2fy + g2 + f2 = g2 + f2 - csin2α - (g2 + f2)cos2 α 
(x+g)2 + (x+f)2 = (√(g2 + f2 - csin2α - (g2 + f2)cos2 α ))2

We find that the circles share a common center and differ only in their radii.


From the inherent property of tangents Circle 1 is bigger than Circle 2. It is required to find <

Fact:
If two chords intersect in the interior of a circle, then the measure of each angle is one half the sum of themeasures of the arcs intercepted by the angle and its vertical angle.

If a tangent and a secant intersect in the exterior of a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs. 

If two tangents intersect in the exterior of a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs.

If two secants intersect in the exterior of a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs.

Derivation
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