A unit circle has its centre at (a, 0). Find the number of points of intersection of this circle with the curve x² = y²
Answers
Answer:
When |a| = √2, the curve x²=y² meets the circle in two points.
When |a| < √2, the curve x²=y² meets the circle in four points.
When |a| > √2, the curve x²=y² does not meet the circle.
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Step-by-step explanation:
x² = y² just describes the pair of lines x = y and x = -y.
Since the circle is centred on the x-axis, each of these lines meets the circle the same number of times, so the answer is just 2 times the number of points of intersection with the line y = x.
The unit circle and the line y = x has just one point of intersection
<=> the circle is tangent to the line
<=> the origin, the point of tangency and the centre (a, 0) form an isosceles right angled triangle with legs of length 1, so hypotenuse of length √2
<=> a = ± √2
So when |a| = √2, the line y=x meets the circle once.
When |a| < √2, the line y=x meets the circle twice.
When |a| > √2, the line does not meet the circle.
Remembering that x²=y² is a pair of lines, we double this to get:
When |a| = √2, the curve x²=y² meets the circle in two points.
When |a| < √2, the curve x²=y² meets the circle in four points.
When |a| > √2, the curve x²=y² does not meet the circle.