### Video Transcript

Convert the parametric equations π₯ equals ln of a half π‘ and π¦ equals three π‘ squared to rectangular form.

In order to convert these parametric equations into rectangular form, we need to eliminate the variable π‘. So, weβre left with an equation containing π¦ and π₯ only. Letβs consider the equation. [π₯] is equal to ln of a half π‘ and rearrange it to make π‘ the subject. To eliminate the ln from the right-hand side, we put both sides as a power of π. On the right-hand side, the π and ln cancel so that π to the power of π₯ is equal to a half π‘. Our final step is to multiply both sides by two. Two π to the power of π₯ is equal to π‘.

Letβs now consider our second equation π¦ equals three π‘ squared. As π‘ is equal to two π to the power of π₯, we can substitute this into the equation. This gives us three multiplied by two π to the power of π₯ squared. Two π to the power of π₯ multiplied by two π to the power of π₯ is equal to four π to the power of two π₯. Remember, when multiplying, we can add our exponents or indices.

Our equation simplifies to π¦ is equal to three multiplied by four π to the power of two π₯. Three multiplied by four is equal to 12. So, π¦ is equal to 12π to the power of two π₯. We now have our equation in rectangular form by eliminating the variable π‘ and leaving our final equation in terms of π¦ and π₯.