Find the equation of the line passing through the vertices of this curve=%5Cfrac%7B-2x-9%7D%7Bx+5%7D "f(x)=\frac{-2x-9}{x+5}")
.
Solution
Let the vertices be 
and 
where 
.
∴&space;&space;%5Cfrac%7B-2a-9%7D%7Ba+5%7D "f (a) = \frac{-2a-9}{a+5}")
and &space;&space;%5Cfrac%7B-2b-9%7D%7Bb+5%7D "f (b) = \frac{-2b-9}{b+5}")
The gradient of tangent to the curve is&space;=%5Cfrac%7B1%7D%7B%7B%7B(x+5)%7D%5E%7B2%7D%7D%7D. "f ' (x) =\frac{1}{{{(x+5)}^{2}}}.")
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) 
and where . 
∴
The gradient of tangent to the curve is
∴
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) 
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