l.VJ 



LLFI7C 9MOMKTKY 



(-1 



[CM. 





= V + y\ H fyi + ft 



i.., ffo square of the length of the tangent from a // 

 external point to the circle z 2 + y 2 + 2Gx + 2Fy +(7=0* 

 t't obtained by writing (he first member only of this equa 

 tulstituting in it the coordinates of the given point.} 



89. From any point outside of a circle two tangents to the 

 circle can be drawn, (a) Let the equation of the circle be 



^ + y2 = r 2, (1) 



then (Art. 83) the line 



y = mx + r Vl + 7W 2 ... (2) 



is, for all values of m, tangent to this circle. Let P l = (r v y^) 

 l>e any given point outside the circle (1); then tin- tan _:<> it 

 (2) will pass through P l if, and only if, m be given a value 

 such that the equation 



y l = mx 1 + rVl 4- w a . . (3) 



shall be satisfied. 



Transposing, squaring, and rearranging equation (- *), it 

 is clear that it will be satisfied if, and only if, m is given a 

 value such that the equation 



(r 2 -ar^m* + Sa-^m + r> - y* = 

 is satisfied; t.e., equation (:',) is satisfied if, and only if, 



m = 



- g^t 



4- y t - 



(4) 



Equation (4) gives <M?O, and only two, real values for m 

 (> p y t ) is outside of the circle, for then r^ +y? r* is 



If the circle is given by the equation A& + Aj/ 2 + 2Gx + 2Fy+C = 0, 

 it must first be divided by A before applying this theorem. 



t The expression *i a + yi* + 2 Oxi + 2 Jfy + C Is called the power o/ the 

 point PI = (*!, yi) with regard to the circle x 2 + y 8 + 2 (/z -f 2 /^ + C = 0. 



