# The graph of y= `(ax+b)/(cx+d)` where a, b, c and d are non-zero constants intersects the y axis at the point where y=1 and the x axis at x=-0.5. At the point it intersects the y axis it is also...

The graph of y= `(ax+b)/(cx+d)`

where a, b, c and d are non-zero constants intersects the y axis at the point where y=1 and the x axis at x=-0.5. At the point it intersects the y axis it is also parallel to the line y=3x+10. Find the values of a,b,c and d. (hint: your answer may not necessarily be a number.

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### 1 Answer

Given that the graph of `y=(ax+b)/(cx+d)` contains the points (0,1),(-1/2,0) and the slope of the tangent line at (0,1) is 3, find a,b,c, and d.

(1) If x=0 then `y=b/d` ; y=1 ==> b=d

(2) If `x=-1/2` then `y=(-a/2+b)/(-c/2+d)=(-a/2+b)/(-c/2+b)`

y=0==> `-a/2+b=0, -c/2+b !=0` . `-a/2+b=0==>b=a/2`

(3) Using the quotient rule we find:

`y'=((cx+a/2)a-(ax+a/2)c)/((cx+a/2)^2)` When `x=0,y'=3` so

`3=(a^2/2-(ac)/2)/((a/2)^2)` ==>`(3a^2)/4=(a(a-c))/2` ==> `(3a)/2=a-c` ==>`c=-a/2`

Thus given `a!=0` ; `b=a/2,c=-a/2,d=a/2`

Example graph: if a=2 then we have `y=(2x+1)/(-x+1)` :

with the tangent line in red.