Question

For which values of a and b is the line b 2x + y = tangent to the parabola 2

y = ax

when x = 2? 

Answer 1

$\text{The equation of the parabola :- } y=ax^2\\\;\\ \therefore \text{Slope of the parabola at point x=2 ,}\\ \left(\dfrac{dy}{dx}\right)_{x=2}=\left(2ax\right)_{x=2}=2a\times 2=4a$

$\text{Now equation of the line : }\\ 2x+y=b\\ \Rightarrow y=-2x+b\\ \;\\ \text{So the slope of the line }=-2\\\;\\ \text{Now equating slopes, we get,}\\ 4a=-2\\ \text{or, } a=-\dfrac{1}{2}$

$\text{Now, at }x=2,\\ \text{Ordinate of parabola} = a\times2^2=\dfrac{-1}{2}\times2^2=-2\\ \text{and Ordinate of the line}=-2\times 2+b=b-4\\\;\\ \text{Now equating the ordinates, we get,}\\ b-4=-2\\ \text{or, }b=2\\\;\\ \therefore \text{For } a=-\dfrac{1}{2}\;\;\text{and}\;b=2, \;\;\;\; \text{2x+y=b is a tangent of $y=ax^2$.}$

So,

$$\fcolorbox{black}{yellow}{\textcolor{blue}{$ a=-\dfrac{1}{2},\;b=2$}}$$