Convert the parametric equations 𝑥 equals ln of a half 𝑡 and 𝑦 equals three 𝑡 squared to rectangular form.
In order to convert these parametric equations into rectangular form, we need to eliminate the variable 𝑡. So, we’re left with an equation containing 𝑦 and 𝑥 only. Let’s consider the equation. [𝑥] is equal to ln of a half 𝑡 and rearrange it to make 𝑡 the subject. To eliminate the ln from the right-hand side, we put both sides as a power of 𝑒. On the right-hand side, the 𝑒 and ln cancel so that 𝑒 to the power of 𝑥 is equal to a half 𝑡. Our final step is to multiply both sides by two. Two 𝑒 to the power of 𝑥 is equal to 𝑡.
Let’s now consider our second equation 𝑦 equals three 𝑡 squared. As 𝑡 is equal to two 𝑒 to the power of 𝑥, we can substitute this into the equation. This gives us three multiplied by two 𝑒 to the power of 𝑥 squared. Two 𝑒 to the power of 𝑥 multiplied by two 𝑒 to the power of 𝑥 is equal to four 𝑒 to the power of two 𝑥. Remember, when multiplying, we can add our exponents or indices.
Our equation simplifies to 𝑦 is equal to three multiplied by four 𝑒 to the power of two 𝑥. Three multiplied by four is equal to 12. So, 𝑦 is equal to 12𝑒 to the power of two 𝑥. We now have our equation in rectangular form by eliminating the variable 𝑡 and leaving our final equation in terms of 𝑦 and 𝑥.