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Question Video: Determining the Position of a Given Point on the 𝑥- and 𝑦-Axes Mathematics

On which of the following does the point (3, 6) lie? [A] the circle 𝑥² + 𝑦² = 225 [B] the 𝑦-axis [C] the straight line 𝑦 = 2𝑥 [D] the 𝑥-axis

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Video Transcript

On which of the following does the point three, six lie? (A) The circle 𝑥 squared plus 𝑦 squared equals 225, (B) the 𝑦-axis, (C) the straight line 𝑦 equals two 𝑥, or (D) the 𝑥-axis.

So if we take a look at our point three, six, then what we know is the three corresponds to the 𝑥-coordinate and the six corresponds to the 𝑦-coordinate. So to help us solve this problem and actually visualize what’s happening, what I’ve done is drawn a sketch of a graph. And I’ve plotted our point three, six. Well, from a quick observation, we can see that it’s not on either of our axes. So therefore, we know it’s not on the 𝑦-axis, and it’s not on the 𝑥-axis.

But we didn’t need to do a sketch to know this. We would have known that it’s on neither of the axes for another reason. Because if we take a look at our coordinates, well, we can see that if it was on the 𝑥-axis, then 𝑦 would be equal to zero. So we’d have to have a zero coordinate for 𝑦. And if it was on the 𝑦-axis, we’d have an 𝑥-coordinate of zero because it’s on the line 𝑥 equals zero. And as we can see that neither of the coordinates are zero, we could say that definitely it’s not on either of our axes.

Okay, so now let’s have a look at (A). Well, for (A), we have the equation of a circle, which is 𝑥 squared plus 𝑦 squared equals 225. Well, to see whether our point lies on this circle, what we’re gonna do is substitute in our 𝑥- and 𝑦-values. And what we’re gonna do is see if they do in fact equal to 225, so the right-hand side of the equation. So we’ve got three squared plus six squared, which is nine plus 36, which is equal to 45. And 45 is not equal to 225. So therefore, we can rule this out because we can say that the point does not lie on the circle 𝑥 squared plus 𝑦 squared equals 225.

So it looks like (C) is gonna be the correct answer. But we can double check by substituting into the equation for the straight line 𝑦 equals two 𝑥. Well, what we get six is equal to two multiplied by three. And that’s because 𝑥 is equal to three and 𝑦 is equal to six. Well, this is correct. So what we can say is the point three, six lies on the straight line 𝑦 equals two 𝑥. So the correct answer is (C).

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