Video: Finding the Position of a Point with Respect to a Circle given Its Coordinates and the Equation of the Circle

Where is the point (4, 2) in relation to circle (𝑥 − 4)² + (𝑦 − 1)² = 1?


Video Transcript

Where is the point four, two in relation to circle 𝑥 minus four all squared plus 𝑦 minus one all squared equals one?

Well, in order to solve this problem, what we need to do is decide whether the point is on inside or outside of our circle. And to do that, what we’re going to do is substitute in our values for 𝑥 and 𝑦. And we could see that in this question, because of the point we’ve got, our 𝑥-value is going to be four and our 𝑦-value is going to be two. So, what we’re gonna do when we substitute in our values four and two for 𝑥 and 𝑦 into the equation of a circle is determine whether this point is inside, outside, or on the circle itself.

And what we know is that if the left-hand side of our circle equation is greater than the right-hand side, so in this case greater than one, then we know that the point is outside of the circle. If the left-hand side is less than one, it’s gonna be inside the circle. However, if the left-hand side is equal to one, so equal to the value on the right-hand side, we know that it’s gonna be on the circle. So, what we’re gonna get is four minus four all squared plus two minus one all squared, which is gonna give us zero squared plus one squared cause four minus four is zero and two minus one is one. So, we’ve got zero squared plus one squared, which is equal to one. So, therefore, it is equal to the right-hand side of our equation.

So, therefore, what we can say is that the point four, two lies on the circle. And we know that because we’ve shown for the left-hand side of the equation when you substitute in 𝑥 equals four and 𝑦 equals two is in fact equal to the right-hand side of the equation, which is our 𝑟 squared, which is equal to one.

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