288 TANGENT AND NORMAL. EXAMPLES. 



2. In the curve y 3 = (x af (xc), shew that the tangent 

 is parallel to the axis of x at the point for which 



3. In the curve x*y* = a 3 (x*\- y}, the tangent at the origin is 



inclined at an angle of 135 to the axis of a:. 



4. In the curve x 2 (x + y) = a 2 (x y), the equation to the 



tangent at the origin is y = x. 



5. In the curve x% + y * = a^ find the length of the perpen- 



dicular from the origin on the tangent at (x, y) ; also 

 find the length of that part of the tangent which is 

 intercepted between the two axes. 



Results. (1) U(aayy)\ (2) a. 



6. If c x , y l} be the parts of the axes of x and y intercepted 



by the tangent at the point (x, y} to the curve 



7. Shew that all the curves represented by the equation 



V-* 



different values being assigned to n, touch each other 

 at the point (a, b). 



8. In the curve y n = a?~ l x, express the equation to the 



tangent in its simplest form ; and determine the value 

 of n when the area included between the tangent and 

 the co-ordinate axes is constant. 



9. If the normal to the curve x% + y% = a$, make an angle <f> 



with the axis of x, shew that its equation is 



y cos < x sin <f> = a cos 2<. 



