Question

If the parabola $y=a{x}^2-6x+b$ passes through $(0,2)$ and has its tangent at $x=\frac{3}{2}$ parallel to the $x-$axis then

• A. $a=2,b=-2$
• B. $a=2,b=2$
• C. $a=-2,b=2$
• D. $a=-2,b=-2$

Answer 1

Answer: (B)

$a=2,b=2$

Given equation of parabola is
$y=a{x}^2-6x+b$
Since it passes through $(0,2)$
$2=0+b$
$\Rightarrow b=2$
Also, $\frac{dy}{dx}=2ax-6$
So, slope of tangent at $(x=\frac{3}{2})=2a\left(\frac{3}{2}\right)-6$
$=3a-6$
Since the tangent at $x=\frac{3}{2}$ is parallel to $x-$axis
$\Rightarrow 3a-6=0$
$\Rightarrow a=2$
Hence, $a=2,b=2$