A circular pond has the equation x² + y² - 6x - 8y = 0, so it passes through the origin, with…
Contributed by StemPy LabTry it on paper first. A hint, the key idea and the worked solution are here when you need them.
A circular pond has the equation , so it passes through the origin, with lengths in metres. Ropes are tied at the origin and stretched across the pond to its edge, in every direction. The midpoints of these ropes form a curve. Find the centre of that curve and its radius.
Need a reminder? The key ideaShow
From an outside point the tangent has length , the chord of contact is , and the area of the triangle is where is the tangent length.
The worked solution
Have you tried it?
Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.
Step 1. The circle. The centre is and the radius is metres.
Step 2. The condition. A line from the centre to the midpoint of a chord is perpendicular to the chord. The chord is along . So :
Step 3. The locus. Replace by : . This is a circle with centre and radius metres.
Answer: the midpoints lie on a circle with centre and radius metres. It is the circle on as a diameter. That also follows from Step 2, since the angle is a right angle.
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