This lesson will cover a few examples illustrating relative position of two circles. I’ll take up one example for each case, and won’t be showing all the calculations in much detail. Go back to the previous lesson for the concept explanation, and head over here in case you forgot how to find the centre or the radius of a circle.

**Example 1 **Determine the relative position of the two circles x^{2} + y^{2 }= 1 and x^{2} + y^{2} – 6x – 8y + 16 = 0.

**Solution** Here, the centres of the two circles are C_{1}(0, 0) and C_{2}(3, 4) respectively, making C_{1}C_{2} = 5. Next, the radii r_{1} and r_{2} are 1 and 3 respectively.

As C_{1}C_{2} > r_{1} + r_{2}, the two circles lie **outside each other**.

**Example 2 **Determine the relative position of the two circles x^{2} + y^{2} – 6x + 8 = 0 and x^{2} + y^{2} + 2x – 8 = 0.

**Solution **In this case, the centres of the two circles are C_{1}(3, 0) and C_{2}(-1, 0) respectively, which means C_{1}C_{2} = 4. The radii of the circles, r_{1} and r_{2}, are 1 and 3 respectively.

This makes C_{1}C_{2} = r_{1} + r_{2}, implying that the two circles lie **touch externally.**

**Example 3 **Determine the relative position of the two circles x^{2} + y^{2} = 16 and x^{2} + y^{2} – 8x – 6y = 0

**Solution** In this case, the centres of the two circles are C_{1}(0, 0) and C_{2}(4, 3) respectively, hence C_{1}C_{2} = 5. The radii of the circles, r_{1} and r_{2}, are 4 and 5 respectively.

Now, C_{1}C_{2} < r_{1} + r_{2}. But this could mean anything – intersecting, touching internally, or one lying inside the other. So, whenever you find C_{1}C_{2} to be smaller r_{1} + r_{2}, you must compare it with r_{1} – r_{2} as well.

Here C_{1}C_{2} is also greater than r_{1} – r_{2}, which means that the two circles **intersect at two points**.

**Example 4 **Determine the relative position of the two circles x^{2} + y^{2} = 16 and x^{2} + y^{2} – 6x + 8 = 0

**Solution** Here, C_{1} is (0, 0) and C_{2} is (3, 0), which makes C_{1}C_{2} = 3. Further, r_{1} = 4 and r_{2} = 1. We can see that C_{1}C_{2} = r_{1} – r_{2} , which means the circles **touch internally.**

**Example 5 **Determine the relative position of the two circles x^{2} + y^{2} = 25 and x^{2} + y^{2} – 2x – 4y + 4 = 0

**Solution** Here, C_{1} is (0, 0) and C_{2} is (1, 2), which makes C_{1}C_{2} = \( \sqrt{5} \) . Further, r_{1} = 5 and r_{2} = 1. We can see that C_{1}C_{2} < r_{1} + r_{2}, but cannot conclude anything yet. We must compare C_{1}C_{2} with r_{1} – r_{2} as well.

As C_{1}C_{2} < r_{1} – r_{2}, the circle with centre C_{2} **lies completely inside the other.**

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And that should cover everything about relative position of two circles. In the next lesson, I’ll talk about common tangents to two circles.