If (2,6) lies on the curve \(\displaystyle{f{{\left({x}\right)}}}={a}{x}^{{{2}}}+{b}{x}\)

Brielle James

Brielle James

Answered question

2022-04-06

If (2,6) lies on the curve f(x)=ax2+bx and y=x+4 is a tangent to the curve at that point. Find a and b?

Answer & Explanation

Buizzae77t

Buizzae77t

Beginner2022-04-07Added 13 answers

The gradient of the tangent to a curve at any particular point is given by the derivative of the curve at that point.
We have:
f(x)=ax2+bx
We are given that the point (2,6) lies on the curve:
f(2)=6
4a+2b=6
2a+b=3...[A]
If we differentiate the parabola equation, then we have:
f'(x)=2ax+b
The gradient of the tangent at (2,6) is given by:
mT=f(2)=4a+b
We also know that the tangent equation is y=x+4 and so comparing with the standard straight line equation y=mx+c:
mT=1
4a+b=1...[B]
We now solve the equations [A] and [B] simultaneous:
[B]-[A]2a=2a=1
Substitute a=-1 into [A]:
b2=3b=5
Hence we have:
a=-1, b=5

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