Finding coefficients of quadratic formula given certain properties

Question

Given the quadratic equation $ax^2+bx+c$ , how do you find $a,b$ and $c$ given you know:

the gradient of the curve at the $y$ intercept

the equation of the tangent at point $P$

the gradient of the normal at point $P$

I haven’t included the specific equations and stuff as I would like to work it out myself, I just need to know what steps to take.

Answer 1

Hint. In other words, you want to determine the curve $y=ax^2+bx+c,$ given that

(1) $y'=m$ at $x=0,$

(2) $y'=n$ at $x=p,$

(3) $y=q$ when $x=p.$

You have three linear equations in $a,b$ and $c.$

PS. The value $n$ is equal to $-1/n,$ the gradient of the normal at $P(p,q).$

Can you continue now?

Answer 2

Let $f(x) = ax^2 + bx + c$ be the curve $C$ in question.

If we know the gradient of $C$ at the $y$-intercept (i.e. where $x=0$) is $m_0$, then that is the same as saying we know that $f'(0) = m_0$.

If we know the equation of the line $L$ which is tangent to $C$ at the point $P$ with co-ordinates $(x_P, y_P)$, then we know two things:

1. $P$ lies on $C$, and thus we know that $f(x_P) = y_P$; and
2. the equation describing $L$ can be written in the form $y = m_P x + d$, whence $m_P$ is the gradient of this line. Thus, we know that $f'(x_P) = m_P$.

Knowing the gradient of the normal to $C$ at $P$ provides no extra information, since this is guaranteed to be equal to $\frac{-1}{m_P}$.

You can substitute $f(x)$ and $f'(x)$ into the three "we know that" statements given above, and use the resulting system of three linear equations to solve for $a$, $b$, and $c$.