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

To determine the values of \(a\), \(b\), and \(c\) for the quadratic curve \(y = ax^2 + bx + c\) given the specified conditions, we proceed as follows:

1. Substitute the point \((1, 2)\) into the curve equation since the curve passes through this point:
   • \(y = ax^2 + bx + c \) results in \(2 = a(1)^2 + b(1) + c\).
   • This simplifies to: \(a + b + c = 2\). (1)
2. The tangent at the origin is \(y = x\), implying that the derivative of the curve at \(x = 0\) should equal 1 (the slope of the tangent line):
   • Differentiating \(y = ax^2 + bx + c\) gives \(\frac{dy}{dx} = 2ax + b\).
   • At the origin \(x = 0\): \( \frac{dy}{dx} = b = 1 \). (2)
3. Also, since the curve passes through the origin \((0,0)\), substitute \((0, 0)\) into the curve:
   • \(0 = a(0)^2 + b(0) + c\) implies \(\ c = 0 \). (3)
4. Substitute \(b = 1\) and \(c = 0\) into equation (1) derived from substituting the given point:
   • Equation from step 1: \(a + b + c = 2\).
   • Substituting known values gives: \(a + 1 + 0 = 2\).
   • This simplifies to: \(a = 1\).

From the steps above, we deduce that the values are \(a = 1\), \(b = 1\), and \(c = 0\). Thus, the correct option is: a = 1, b = 1, c = 0.