Video Transcript
The ratio 𝑥 to four equals four to 𝑦, so four is the geometric mean of 𝑥 and 𝑦. Find the geometric mean of 𝑥 plus one over 𝑦 and 𝑦 plus one over 𝑥.
Let’s begin by recalling the definition of a geometric mean. The geometric mean of two numbers 𝑎 and 𝑏, which must have the same sign, is the square root of 𝑎𝑏. We can only find the geometric mean of two numbers which have the same sign because if the two numbers were of opposite signs, then the product would be negative. And the square root of a negative number gives a nonreal result. We are told that four is the geometric mean of 𝑥 and 𝑦, so we know that the square root of the product 𝑥𝑦 is equal to four. Let’s also just explore this ratio for a moment. We are told that the ratio 𝑥 to four is the same as the ratio four to 𝑦.
If these two ratios are the same, then if we divide the value on the left of each ratio by the value on the right, we get the same result. So we have an equation, 𝑥 over four is equal to four over 𝑦. Multiplying both sides of this equation by the two denominators of four and 𝑦, we have 𝑥𝑦 is equal to four squared. Then taking the positive square root of each side of this equation, we have that the square root of 𝑥𝑦 is equal to four. And so we see that four is indeed the geometric mean of 𝑥 and 𝑦.
Now, we are asked to find the geometric mean of two other quantities: 𝑥 plus one over 𝑦 and 𝑦 plus one over 𝑥. Provided these two numbers are of the same sign then, their geometric mean is the square root of their product. It’s the square root of 𝑥 plus one over 𝑦 multiplied by 𝑦 plus one over 𝑥. We can deduce that these two quantities do have the same sign because 𝑥 and 𝑦 have the same sign. If 𝑥 and 𝑦 are both positive, then every term involved in this expression is positive. And so we’re multiplying one positive value by another positive value. If, on the other hand, 𝑥 and 𝑦 are both negative, then one over 𝑦 and one over 𝑥 are also both negative. So everything in the expression is negative, and we’re multiplying two negative values together.
Let’s consider then how we can manipulate this expression, and we’ll begin by distributing the parentheses. Using the FOIL method, we have 𝑥𝑦 plus 𝑥 multiplied by one over 𝑥 plus one over 𝑦 multiplied by 𝑦 plus one over 𝑥 multiplied by one over 𝑦. Each of the terms in the center of this expansion simplify to one. And one over 𝑥 multiplied by one over 𝑦 is one over 𝑥𝑦. So we have 𝑥𝑦 plus one plus one plus one over 𝑥𝑦, which simplifies to 𝑥𝑦 plus two plus one over 𝑥𝑦. The geometric mean of these two expressions then is the square root of 𝑥𝑦 plus two plus one over 𝑥𝑦.
Now next, we recall that we know the geometric mean of 𝑥 and 𝑦, the square root of 𝑥𝑦 is equal to four. Squaring both sides of this equation tells us that 𝑥𝑦 is equal to 16. And we can now substitute this value for 𝑥𝑦 in two places in our expression for the geometric mean, which gives the square root of 16 plus two plus one over 16. That’s the square root of 18 and one sixteenth or the square root of 18.0625. And we can then use the calculator to evaluate this, and it gives 4.25.
So given that the geometric mean of 𝑥 and 𝑦 is four, we found that the geometric mean of 𝑥 plus one over 𝑦 and 𝑦 plus one over 𝑥 is 4.25.
