Question

If the line y = 2x touches the curve y = ax^2 + bx + c at the point where x = 1 and the curve passes through the point (-1, 0), then the values of a, b, c are

• A. $a=\frac{1}{2}b=1,c=\frac{1}{2}$
• B. $a=1,b=\frac{1}{2}c=\frac{1}{2}$
• C. $a=\frac{1}{2},b=\frac{1}{2},c=1$
• D. $a=1,b=2,c=-1$

Answer 1

The correct option is A

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

The given curve is $y=a{x}^2+bx+c$ ...(1)
Since the point (-1, 0) lie on it.
∴ a - b + c = 0 ...(2)
Also, y = 2x is a tangent to (1) at x = 1, so that y = 2,
Since the point (1, 2) lies on equation (1), ...(3)
$\therefore$ a + b + c =2
Also ${\left[\frac{dy}{dx}\right]}_{(1,2)}=(2ax+b{)}_{12}=2.\therefore 2a+b=\mathrm{2.....}(4)$
On solving equation (2), (3) and (4), we get
$a=\frac{1}{2},b=1,c=\frac{1}{2}$
Hence (a) is the correct answer