### Video Transcript

A curve has parametric equations π₯ is equal to seven π cubed plus five π squared plus π plus four and π¦ is equal to six π squared minus six π minus eight. Find π for which the tangent is horizontal.

In this question, we are given parametric equations where our coordinates π₯ and π¦ are given in terms of π. We know that when the tangent is horizontal, its gradient or slope, dπ¦ by dπ₯, must be equal to zero. We also know that dπ¦ by dπ₯ is equal to dπ¦ by dπ divided by dπ₯ by dπ. This can also be written using the chain rule as dπ¦ by dπ multiplied by dπ by dπ₯. We multiply dπ¦ by dπ by the reciprocal of dπ₯ by dπ.

Letβs begin by considering π₯, which is equal to seven π cubed plus five π squared plus π plus four. We can differentiate this term by term to get an expression for dπ₯ by dπ. Differentiating seven π cubed gives us 21π squared, differentiating five π squared gives us 10π, and differentiating π gives us one. When we differentiate the constant four, we get zero. Therefore, dπ₯ by dπ is equal to 21π squared plus 10π plus one.

We can repeat this process for π¦, which is equal to six π squared minus six π minus eight. dπ¦ by dπ is equal to 12π minus six. We can then find an expression for dπ¦ by dπ₯ by dividing dπ¦ by dπ by dπ₯ by dπ. dπ¦ by dπ₯ is equal to 12π minus six divided by 21π squared plus 10π plus one.

For the tangent to be horizontal, we know that dπ¦ by dπ₯ must equal zero. This means that we need to solve this equation equal to zero. As the denominator cannot be equal to zero, the numerator must be. 12π minus six must equal zero. Adding six to both sides gives us 12π is equal to six. Dividing both sides of this equation by 12 gives us π is equal to six over 12 or six twelfths. This fraction is equivalent to one-half. The value of π for which the tangent is horizontal is one-half.

Whilst we donβt need to in this question, we could then substitute this value for π back into our equations for π₯ and π¦. This would give us the coordinate of the point at which the tangent is horizontal.