Problem Study (Analytic Geometry)

Find the equation of the line passing through the vertices of this curve.
Solution
   
Let the vertices be and where .  

and    
The gradient of tangent to the curve is  

At vertices, the tangents are parallel. 
 
 
 
 
 
 
Since
and .
 
Since the line passing through vertices is perpendicular to the respective tangents, its gradient is  .
Hence  
            
            
            
            
           (or)  
        (or)
       (or) and
          (or)
Therefore, the vertices are (– 4, – 1) and (– 6, – 3).
Hence the equation of required line is

  
(or)