Question

A curve is described by 𝑦=𝑝𝑥²+𝑞𝑥, where 𝑝 and 𝑞 are constants.

a. Find an expression for the gradient of this curve at any point.

b. Given that at the point (1,−2) the gradient is 6, calculate the values of 𝑝 and 𝑞.

c. Show that the equation of the normal to the curve at the point (1,−2) can be written as 𝑥+6𝑦+11=0.

Answer 1

a.

$$y=px^2+qx$$

$$grad (y)=(px^2+qx)'=2px+q$$

b.

$$-2=p(1)^2+q(1)$$ $$2p(1)+q=6$$

$$q=-2-p$$ $$2p-2-p=6$$

$$p=8, q=-10$$

c.

Find the slope of the normal at the point $(1, -2)$

$$slope=m=-\dfrac{1}{grad (y)}=-\dfrac{1}{6}$$

The equation of the normal at the point $(1,-2)$ is

$$y-(-2)=-\dfrac{1}{6}(x-1)$$

$$6y+12=-x+1$$

Then he equation of the normal to the curve at the point $(1,−2)$ can be written as

$$x+6y+11=0$$