Question

If the curve $y=a x^{2}+b x+c, x \in R,$ passes through the point (1,2) and the tangent line to this curve at origin is $y = x ,$ then the possible values of $a , b , c$ are :

• A. $a =\frac{1}{2}, b =\frac{1}{2}, c =1$
• B. $a =1, b =0, c =1$
• C. $a =1, b =1, c =0$
• D. $a=-1, b=1, c=1$

Answer 1

The Correct Option is C

Given the quadratic curve \(y = ax^2 + bx + c\), we know it passes through the point (1, 2) and has a tangent line at the origin with the equation \(y = x\). We need to determine the values of \(a\), \(b\), and \(c\).

1. Since the curve passes through the point (1, 2), we substitute these values into the equation of the curve: \(2 = a(1)^2 + b(1) + c = a + b + c\) (Equation 1)
2. The tangent at the origin is \(y = x\), which implies that the derivative of the curve at the origin should equal 1 (the slope of the line \(y = x\)). Differentiating the curve equation 

\[y = ax^2 + bx + c\]

1. , we get: \(\frac{dy}{dx} = 2ax + b\) Evaluate this at \(x = 0\): \(b = 1\) (Equation 2)
2. Substitute the value of \(b = 1\) from Equation 2 back into Equation 1: \(2 = a + 1 + c \rightarrow a + c = 1\) (Equation 3)
3. Now, evaluate the options using Equations 2 and 3 to find which set of values satisfies both equations:
   • Option A: \(a = \frac{1}{2}, b = \frac{1}{2}, c = 1\)
     Substituting into the equations: Equation 2 is not satisfied since \(b \neq 1\). Hence, Option A is incorrect.
   • Option B: \(a = 1, b = 0, c = 1\)
     Substituting into the equations: Equation 2 is not satisfied since \(b \neq 1\). Hence, Option B is incorrect.
   • Option C: \(a = 1, b = 1, c = 0\)
     Substituting into the equations: Equation 1: \(2 = 1 + 1 + 0 = 2\) ✅
     Equation 2: \(b = 1\) ✅
     Equation 3: \(1 + 0 = 1\) ✅
     This option satisfies all conditions.
   • Option D: \(a = -1, b = 1, c = 1\)
     Substituting into the equations: Equation 3 is not satisfied since \(-1 + 1 = 0\). Hence, Option D is incorrect.

Thus, the correct values of \(a\), \(b\), and \(c\) that satisfy all the conditions are given in Option C: \(a = 1, b = 1, c = 0\).