Video: EG19M2-ALGANDGEO-Q06

EG19M2-ALGANDGEO-Q06

04:14

Video Transcript

If π₯ plus π¦π is equal to π plus ππ over π minus ππ, find π₯ squared plus π¦ squared.

To find the value of π₯ and π¦, weβre going to need to begin by evaluating this expression on the right-hand side of our equation. Here, we have the quotient of two complex numbers. To divide these complex numbers, weβd multiply both parts, essentially the numerator and the denominator, by the conjugate of the denominator. The conjugate of π minus ππ is π plus ππ. Remember we simply change the sign between the two terms. Weβre therefore going to multiply π plus ππ and π minus ππ by π plus ππ. Weβll expand or distribute these brackets as normal.

Letβs look at the FOIL method for π plus ππ multiplied by π plus ππ. The F part of FOIL stands for first. We multiply the first term in the first bracket by the first term in the second bracket. π multiplied by π is π squared. O stands for outer. We multiply the outer two terms. Thatβs π multiplied by ππ, which is πππ. We then multiply the inner two terms which is once again πππ. And the L stands for last. Weβre going to multiply the last term in each of the brackets. And doing so, we get π squared π squared. Remember π is the square root of negative one. So π squared is negative one and π squared π squared is equal to negative π squared. Collecting like terms and the expression simplifies to π squared plus two πππ minus π squared.

Weβll now multiply the denominator by π plus ππ. This time, we get π squared plus πππ minus πππ minus π squared π squared. πππ minus πππ is zero and negative π squared π squared becomes negative π squared multiplied by negative one which is simply π squared. And weβve gone some way to dividing π plus ππ by π minus ππ. Remember we want to write this in the form π₯ plus π¦π. So weβre going to need to separate the real and complex components. And when we do, we get π squared minus π squared over π squared plus π squared plus two πππ over π squared plus π squared.

Remember π₯ represents the real component of our complex number. Itβs π squared minus π squared over π squared plus π squared and π¦ represents the imaginary component. Here, itβs the coefficient of π. So itβs two ππ over π squared plus π squared. Now that we have expressions for π₯ and π¦, we can work out the value of π₯ squared plus π¦ squared. To square each of these fractions, we square the numerators and denominators individually.

Remember we do need to be a little bit careful. We canβt just square each of the individual terms. We need to expand these brackets like we did before. And when we do, we get the numerator to be π to the power of four minus two π squared π squared plus π to the power of four. Weβll leave the denominator as it is because it will help us simplify the expression at the end. And π¦ squared is equal to four π squared π squared all over π squared plus π squared all squared.

We can add these two expressions by simply adding their numerators. And since negative two π squared π squared plus four π squared π squared is two π squared π squared, we get that π₯ squared plus π¦ squared is equal to π to the power of four plus two π squared π squared plus π to the power of four all over π squared plus π squared squared. Now in fact, if we refer to the numerator of this expression we see itβs very similar to the expansion of π squared minus π squared all squared.

In fact, we can then factorise this expression. Itβs π squared plus π squared all squared. And that means π₯ squared plus π¦ squared is equal to π squared plus π squared squared over π squared plus π squared squared, which is of course one. So for this example, π₯ squared plus π¦ squared is equal to one.