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 :
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 :
Updated On: Apr 1, 2026
- $a =\frac{1}{2}, b =\frac{1}{2}, c =1$
- $a =1, b =0, c =1$
- $a =1, b =1, c =0$
- $a=-1, b=1, c=1$
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The Correct Option is C
Solution and Explanation
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 \(c\).
- 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)
- 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
- , we get: \(\frac{dy}{dx} = 2ax + b\) Evaluate this at \(x = 0\): \(b = 1\) (Equation 2)
- Substitute the value of \(b = 1\) from Equation 2 back into Equation 1: \(2 = a + 1 + c \rightarrow a + c = 1\) (Equation 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.
- Option A: \(a = \frac{1}{2}, b = \frac{1}{2}, c = 1\)
Thus, the correct values of \(a\), \(b\), and \(c\) that satisfy all the conditions are given in Option C: \(a = 1, b = 1, c = 0\).
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Concepts Used:
Applications of Integrals
There are distinct applications of integrals, out of which some are as follows:
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Integrals are used to find:
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In Physics
Integrals are used to find:
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