Question

If at $x=1,y=2x$ is tangent to the parabola $y=a{x}^2+bx+c$, then respective values of $a,b,c$ are

• A. $\frac{1}{2},1,\frac{1}{2}$
• B. $1,\frac{1}{2},1/2$
• C. $1,1,1$
• D. none of these

Answer 1

Answer: (A)

$\frac{1}{2},1,\frac{1}{2}$

If at $x=1$ , $y=2x$ is a tangent to parabola $y=a{x}^2+bx+c$

The respective value of $a,b,c$ are:

At $x=1,y=2$

$2=a+b+c.............\left(1\right)$

Equation of tangent at $(1,2)$ is

$\frac{y}{2}+\frac{2}{2}=ax+\frac{b}{2}x+\frac{b}{2}+cy=(2a+b)x+(b+2c-2)$

Comparing with $y=2x$

$2a+b=\mathrm{2...........}\left(2\right)$

$b+2c-2=\mathrm{0.........}\left(3\right)$

Solve $\left(1\right)\left(2\right)\left(3\right)$

We get

$a=\frac{1}{2}b=1c=\frac{1}{2}$

Hence $a,b,c$ are $\frac{1}{2},1,\frac{1}{2}$