i-j:t-u4 . nit: "W f] < y/o.vj -"'> 



14. S* + 4 j* 16, at the point (2. -1). 



124. Equation of a tangent, and of a normal, that paas throofh a 

 liven point which b not oa the cook. 



.< method to be followed in finding the equation of a tangent, or of 

 a normal, that paasea through a given point which b not on la* comic, 

 may l illustrated by the following example; the name method b appli- 

 cable to any conic whatever. 



it t required to find the equation of that tangent to the parabola 



> which paasea through the point (-4, -1). Thb point not being on the 

 parabola, the method of it, assuming for the 



mum.-!.! that it in iossibb to draw such a tangent, l.-t (r,, yj be iU point 

 The equation of thb tangent b (Art. 1 



Since thb tangent passes through the point (I, ~1), therefore equa- 

 (-) b satbfted by the coordinates -4 and -1, 



-fi-M-l+y^ -4 (-4 + x,) -31 =0, . . . (3) 

 ich reduces to r, + y, + 3 = 0. . . (I) 



> furnishes one relation between the two unknown con- 

 stants T I and y, ; another equation between these two unknowns b fur- 

 -d by the fact that (r v y,) b a jH.int on the parabola (1); this 

 equation b 



Solving between equations (4) and (5) gives 



and y, = 



hence, there are two pointa on the given parabola the tangents at which 

 pass through the point ( I, 1); tin ir robrdinates are (- 



; and subatituting either pair 



teee values for x, and y, in equation (2) gives the equation of a 

 straight line that is tangent to the parabola (1), and that passes througn 

 the point ( I. -1). 



So, too, if it b desired to find the equation of a normal through a 

 not on the curve, it b only necessary to anrnme temporarily the COOT* 

 - of the point on the curve through which thb normal passes, and 



