Question

Let `f(x) = (x+2)/(x+3)`. Find the equation(s) of tangent line(s) that pass through a point (0,6)? Sketch the solution?

Let `f(x) = (x+2)/(x+3)`. Find the equation(s) of tangent line(s) that pass through a point (0,6)?

Answer 1

Tangents are `25x-9y+54=0` and `y=x+6`

Explanation:

Let the slope of the tangent be `m`. The equation of tangent then is `y-6=mx` or `y=mx+6`

Now let us see the point of intersection of this tangent and given curve `y=(x+2)/(x+3)`. For this putting `y=mx+6` in this we get

`mx+6=(x+2)/(x+3)` or `(mx+6)(x+3)=x+2`

i.e. `mx^2+3mx+6x+18=x+2`

or `mx^2+(3m+5)x+16=0`

This should give two values of `x` i.e. two points of intersection, but tangent cuts the curve only at one point. Hence if `y=mx+6` is a tangent, we should have only one root for the quadratic equation, which is possible onli if discriminant is `0` i.e.

`(3m+5)^2-4*m*16=0`

or `9m^2+30m+25-64m=0`

or `9m^2-34m+25=0`

i.e. `m=(34+-sqrt(34^2-900))/18`

= `(34+-sqrt256)/18=(34+-16)/18`

i.e. `25/9` or `1`

and hence tangents are `y=25/9x+6` i.e. `25x-9y+54=0`

and `y=x+6`

graph{(25x-9y+54)(x-y+6)(y-(x+2)/(x+3))=0 [-12.58, 7.42, -3.16, 6.84]}