Question:
If line joining the points (0,3) (5, 2) is tangent to curve y= ax/(1 x) then
Answer:
The equation of the line joining the points (0,3) and (5,-2) is
y = -x + 3
The line intersects the curve y = ax/(x + 1) at points whose x-coordinates are given by
-x + 3 = ax/(x + 1)
=> (-x + 3)(x + 1) = ax
=> -x2 + 2x + 3 = ax
=> x2 + (a - 2)x - 3 = 0
The line will be tangent to the curve if the quadratic equation above has a double root, that is when its discriminant equal 0
So, (a - 2)2 - 4 * 1 * (-2) = 0
=> (a - 2)2 + 12 = 0
This is impossible because (a - 2)2 + 12 > 0 for all real number a.
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