Question

Consider the locus of a moving point P(x, y) in the plane which satisfies the condition

2x² = r² + r⁴, where r² = x² + y²

Then, only one of the following statement is true-

• A. For every 0 < r < 1, there are exactly four points on the curve
• B. For every 0 < r £ 1, there are exactly four points on the curve
• C. The locus is a pair of straight lines
• D. None of these

Answer 1

Answer: (A)

For every 0 < r < 1, there are exactly four points on the

curve

Answer 2

We x² £ r² [Q r² = x² + y²]

i.e. 2x² £ 2r²

i.e. r² + r⁴ £ 2r² [Q 2x² = r² + r⁴ given]

i.e. r⁴ – r² £ 0

i.e. r² (r² – 1) £ 0

i.e. 0 £ r² £ 1

i.e. 0 £ r £ 1 [Q r is a +ve quantity]

Also, we can see that the given curve is symmetrical about the X-axis as well as the Y-axis (replacing x by –x or y by –y does not change the equation).

Thus, if (h, k) is a point on the curve then (–h, k), (h, – k) and (–h, –k) are also points on the curve, all of which have the same distance from the origin.

However, there is only one point (0, 0) whose r = 0 and two points (1, 0) and (–1, 0) whose r = 1.

Hence, there are exactly four points on the given curve for every 0 < r < 1.0