# Problem Study (Analytic Geometry)

Find the equation of the line passing through the vertices of this curve$f(x)=\frac{-2x-9}{x+5}$.
Solution
$f(x)=\frac{-2x-9}{x+5}$
Let the vertices be and where .

$f&space;(a)&space;&space;\frac{-2a-9}{a+5}$ and $f&space;(b)&space;&space;\frac{-2b-9}{b+5}$
The gradient of tangent to the curve is $f&space;'&space;(x)&space;=\frac{1}{{{(x+5)}^{2}}}.$

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)