212 



(Alt. 



.1 \ IMT/C GKnMi TRY 



eq, 



~ 



[Cu. VI11. 





If <?=(A, &) be the mid- 

 die point of the < -ln.nl 

 / then 



substituting these values of 

 x l + x 2 and ^ + y 2 * n equa- 

 tion (3), then clearing of 

 fractions and transposing, 

 that equation becomes 



(4) 



But equation (4) asserts that the coordinates (A, k) of 

 the middle point of any one of this system of parallel chords 

 satisfy the equation 



G 



which is therefore the equation of the diameter whose chorda 

 have the slope m. 



EXERCISES 



1. Find the polar of the point (2, 1) with regard to the hyperbola 

 z* - 2 (y* + x) - 4 = 0. Show that this polar passes through (12, 

 and then verify Art. 128, for this particular case, by showing that the 

 polar of (12, 3), with regard to the given hyperbola, passes through ('J. 1 ). 



2. Write the equation of the chord of contact of the tangents drawn 

 through (2, 1) to the hyperbola z 2 - 2y* - 2x - 4=0, then find the 

 points in which it meets the curve, get the equations of the tangei 

 these points, and verify that they pass through the given point (2, 1). 



3. By specializing the coefficients in equation [51], prove that the 

 diameter of a circle is perpendicular to the chords of that diameter. 



