# normal line to the curve

#### mastermin346

A curve $$\displaystyle y=f(x)$$ with the gradient function $$\displaystyle \frac{dy}{dx}=\frac{x^2}{3x-1}.$$

The straight line $$\displaystyle ky=2x-7$$ is normal to the curve $$\displaystyle y=f(x)$$ at $$\displaystyle (2,m)$$.Find the value of $$\displaystyle k$$ and $$\displaystyle m$$.

any help will appreciate,.

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#### mr fantastic

MHF Hall of Fame
The straight line $$\displaystyle ky=2x-7$$ is normal to the curve $$\displaystyle y=f(x)$$ at $$\displaystyle (2,m)$$.Find the value of $$\displaystyle k$$ and $$\displaystyle m$$.

any help will appreciate,.
There is not enough information given. Go back and check the question.

#### mastermin346

There is not enough information given. Go back and check the question.

hi sir..

sorry,my mistake

the question is. A curve $$\displaystyle y=f(x)$$ with gradient function $$\displaystyle \frac{dy}{dx}=\frac{x^2}{3x-1}$$.

#### pickslides

MHF Helper
Well start by substituting $$\displaystyle x=2$$ into $$\displaystyle \frac{dy}{dx}$$ what you you get? What does that mean?

#### mastermin346

Well start by substituting $$\displaystyle x=2$$ into $$\displaystyle \frac{dy}{dx}$$ what you you get? What does that mean?
hi,i get $$\displaystyle \frac{dy}{dx}=\frac{4}{5}$$ then?it mean the gradient right?

#### pickslides

MHF Helper
hi,i get $$\displaystyle \frac{dy}{dx}=\frac{4}{5}$$ then?it mean the gradient right?
Correct, it is the gradient of $$\displaystyle f(x)$$ at $$\displaystyle x=2$$. Now what is the relationship between the gradient and the normal?

Hint: $$\displaystyle m_N\times m_T = -1$$

How can we use this value?

#### mastermin346

Correct, it is the gradient of $$\displaystyle f(x)$$ at $$\displaystyle x=2$$. Now what is the relationship between the gradient and the normal?

Hint: $$\displaystyle m_N\times m_T = -1$$

How can we use this value?
i get the gradient of normal is $$\displaystyle -\frac{5}{4}$$.

#### pickslides

MHF Helper
The straight line $$\displaystyle ky=2x-7$$ is normal to the curve $$\displaystyle y=f(x)$$ at $$\displaystyle (2,m)$$.
so maybe its' time to find $$\displaystyle k$$

Making the form $$\displaystyle y=mx+c$$

$$\displaystyle ky=2x-7\implies y=\frac{2}{k}x-\frac{7}{k}$$

$$\displaystyle \frac{-5}{4}= \frac{2}{k}$$

mastermin346