Question Video: Convert Parametric Equations to Rectangular Form

Convert the parametric equations π‘₯ = ln (1/2 𝑑) and 𝑦 = 3𝑑² to rectangular form.

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Video Transcript

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

In order to convert these parametric equations into rectangular form, we need to eliminate the variable 𝑑. So, we’re left with an equation containing 𝑦 and π‘₯ only. Let’s consider the equation. [π‘₯] is equal to ln of a half 𝑑 and rearrange it to make 𝑑 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 π‘₯ is equal to a half 𝑑. Our final step is to multiply both sides by two. Two 𝑒 to the power of π‘₯ is equal to 𝑑.

Let’s now consider our second equation 𝑦 equals three 𝑑 squared. As 𝑑 is equal to two 𝑒 to the power of π‘₯, we can substitute this into the equation. This gives us three multiplied by two 𝑒 to the power of π‘₯ squared. Two 𝑒 to the power of π‘₯ multiplied by two 𝑒 to the power of π‘₯ is equal to four 𝑒 to the power of two π‘₯. 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 π‘₯. Three multiplied by four is equal to 12. So, 𝑦 is equal to 12𝑒 to the power of two π‘₯. We now have our equation in rectangular form by eliminating the variable 𝑑 and leaving our final equation in terms of 𝑦 and π‘₯.

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