Obtain the Cartesian equation of the curve by eliminating the parameter. x = t 2 , y = ? 7 + t 4...

Question:

Obtain the Cartesian equation of the curve by eliminating the parameter.

{eq}x= t^2, \enspace y=\sqrt{7+t^4}{/eq}

  • A) {eq}y=\sqrt{7+x}{/eq} {eq}\enspace \enspace \enspace {/eq}B) {eq}y=\sqrt{7+x^4}{/eq} {eq}\enspace \enspace \enspace {/eq}C) {eq}y=\sqrt{7+x^2}{/eq} {eq}\enspace \enspace \enspace {/eq}D) {eq}y=\sqrt{7+2x}{/eq}

(Choose the correct one)

Cartesian Equation of a Curve:

We are given two parametric equations {eq}x= f(t) {/eq} and {eq}y= g(t) {/eq} and we need to eliminate the parameter {eq}t. {/eq}

o solve this problem, we need use exponential rules {eq}\sqrt[n]{a^n}= a^{n/n}= a {/eq} and to isolate the parameter {eq}t {/eq} in terms of {eq}x , y {/eq} and equate them. Next, we'll rewrite the expression to choose the correct answer.

Answer and Explanation:

{eq}\displaystyle x =t^2 \ , \ y=\sqrt{7+t^4} {/eq}


Isolate {eq}t {/eq} from the first equation: {eq}x =t^2 {/eq}

{eq}\Rightarrow t =\sqrt x {/eq}


Isolate {eq}t {/eq} from the second equation:

{eq}y=\sqrt{7+t^4} {/eq}

{eq}\Rightarrow 7+t^4= y^2 {/eq}

{eq}\Rightarrow t^4 = y^2-7 {/eq}

{eq}\Rightarrow t= \sqrt[4] {y^2-7} {/eq}

Now equating both values of {eq}t: {/eq}

{eq}\Rightarrow \sqrt{x}= \sqrt[4] {y^2-7} {/eq}

{eq}\Rightarrow (\sqrt{x})^4= y^2-7 {/eq}

{eq}\Rightarrow x^2 = y^2-7 {/eq}

{eq}\Rightarrow y^2 = x^2+7 {/eq}

Isolate y:

{eq}\Rightarrow y= \sqrt{x^2+7} {/eq}


Hence, the Cartesian equation of the curve is {eq}y= \sqrt{7+x^2} {/eq}

ANSWER-C)


Learn more about this topic:

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Evaluating Parametric Equations: Process & Examples

from Precalculus: High School

Chapter 24 / Lesson 3
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