Question

If at $x=1,y=2x$ is tangent to the parabola $y=a{x}^2+bx+c,$ then which of the following option(s) is/are correct?

• A. $a=c$
• B. $a,b,c$ are in A.P
• C. infinite such parabolas exist
• D. $2a+c=2$

Answer 1

Answer: (A), (C)

The correct options are
A $a=c$
C infinite such parabolas exist

From $y=2x$ $x=1$,$\Rightarrow y=2$
Now putting $(1,2)$ in the parabola equation
$2=a+b+c\cdots \left(1\right)$
Equation of tangent at $(x_1,y_1)$ for given parabola is
$\frac{y+y_1}{2}=ax{x}_1+b\frac{(x+x_1)}{2}+c$
putting $(x_1,y_1)\equiv (1,2)$, we get
$y=(2a+b)x+(b+2c-2)$
Comparing with $y=2x$
$2a+b=2...\left(2\right)$
$b+2c-2=0\cdots \left(3\right)$
Solve $\left(1\right),\left(2\right),\left(3\right)$, we get
$c=a,b=2(1-a)$
Hence the parabola will become
$y=a{x}^2+2(1-a)x+a$
Now from here for different values of $a$ except $0$ we will have different parabolas which have tangent as $y=2x$