calculate the calues of a,b,c if the point p(1,2) is on the curve y=ax^2+bx+c and the tangent line if y=x at the point Q(0,0)
To find the values of a, b, and c, given that point P(1,2) is on the curve and the tangent line is y=x at point Q(0,0), we will use the concept of differentiation.
Step 1: Substitute the coordinates of point P(1,2) into the equation of the curve.
2 = a(1^2) + b(1) + c
2 = a + b + c --------------- Equation 1
Step 2: Differentiate the equation of the curve to find the slope of the tangent line.
y = ax^2 + bx + c
dy/dx = 2ax + b
Step 3: Substitute the coordinates of point Q(0,0) into the derivative of the equation of the curve.
0 = 2a(0) + b
b = 0 --------------- Equation 2
Step 4: Substitute the value of b from Equation 2 into Equation 1 to solve for a and c.
2 = a + 0 + c
2 = a + c --------------- Equation 3
Step 5: Substitute the value of a from Equation 3 into Equation 2 to solve for c.
2 = 0 + c
c = 2
Step 6: Substitute the values of b = 0 and c = 2 into Equation 1 to solve for a.
2 = a + 0 + 2
2 = a + 2
a = 0
Therefore, the values of a, b, and c are a = 0, b = 0, and c = 2.