Question

If the curve y = ax² + bx + c passes through (1, 2) and the tangent at origin is y = x, then a, b, c are :

• A. a = 1, b = 1, c = 0
• B. a = 1, b = 0, c = 1
• C. a = -1, b = 1, c = 1
• D. a = 1/2, b = 1/2, c = 1

Hint:

If a curve passes through the origin, the constant term $c$ is always $0$. The coefficient of $x$ ($b$) is the slope of the tangent at the origin.

Answer 1

The Correct Option is A

Step 1: Curve passes through origin $(0, 0)$ because the tangent is defined at the origin. Thus, $c = 0$.
Step 2: Tangent at origin is $y = x \Rightarrow$ Slope $(\frac{dy}{dx})_{x=0} = 1$.
Step 3: $\frac{dy}{dx} = 2ax + b$. At $x=0$, $b = 1$.
Step 4: Curve passes through $(1, 2)$: $2 = a(1)^2 + 1(1) + 0 \Rightarrow a = 1$.
Step 5: Coefficients are $a=1, b=1, c=0$.