ANALYIK gaOMKTIiY [Cn. \. 



But, since each chord parallel to either axis in bisected by the othn, 

 therefore, if (r,, /,) is a point on the curve, then (-r,, + y,) must also 

 be on the curve; 



!>., if - // ,V, + *!>=!, 



thru -1 ./,' - '-'/A',.'/, + -%i* = l, 



and, consequently, H = 0. 



Again. (a', 0) and (0, V) are points on the curve; 



hence =1, Bb'* = 1 ; 



1 R I 



'*" ?"*' &' 



therefore, equation (2) becomes 



-'+*! = !. 

 a" ft" 



This method illustrates how analytic reasoning may often be 

 to shorten or perhaps obviate the algebraic reductions involved in ;i 

 proof. With the similar methods of Arts. 39 and 40, it will su 

 to the reader the power and interest of what are called the modern 

 methods in analytic geometry. 



EXAMPLES ON CHAPTER X 



1. Find the foci, directrices, eccentricity of the ellipse 4* 8 + 3y 2 = 5. 



2. Find the area of the ellipse 4 x 2 + 3y* = 5 (cf. Art. 151, Ex. 9). 



3. Show that the polar of a point on a diameter is parallel to the 

 conjugate diameter. 



4. Find the equations of the normals at the ends of the latus rectum. 

 and prove that each passes through the end of a minor axis if e* + 2 = 1. 



5. Show that the four lines from the foci to two points P l and P s 

 on an ellipse all touch a circle whose center is the pole of P t P r 



6. Tangents are drawn from the point (3, 2) to the ellipse 



z' + 4y = 4. 



Fiml the equation of the line joining (3, 2) to the middle point of the 

 chord of contact. 



7. Find the locus of the center of a circle which passes through the 

 point (0, 3) and touches internally the circle z a + y 2 = '-'">. 



8. Find the length of the major axis of an ellipse whose minor axis 

 is 10, and whose area is equal to that of a circle whose radius is 8. 



