The following (X, Y) points are plotted on X-Y plane and the adjacent points are joined by smooth curve : What will be the nature of curve ?X Y 1 60 2 30 3 20 4 15 5 12
Hyperbola
The shape of a plotted curve can be identified from the relationship between the paired values. A constant difference in Y for equal steps in X gives a straight line, a constant second difference gives a parabola, a constant ratio between successive Y values gives an exponential curve, and a constant product \(XY\) gives a rectangular hyperbola.
Step 1 — Multiply each pair: \(1\times 60=60,\ 2\times 30=60,\ 3\times 20=60,\ 4\times 15=60,\ 5\times 12=60\).
Step 2 — Since \(XY=60\) for every point, the relation is \(Y=\frac{60}{X}\), the equation of a rectangular hyperbola: Y falls steeply at first and then flattens as X grows.
Step 3 — Successive differences in Y are 30, 10, 5, 3, which are not constant, so the points cannot lie on a straight line; the second differences 20, 5, 2 are not constant either, ruling out a parabola; and the ratios \(\frac{30}{60},\frac{20}{30},\frac{15}{20},\frac{12}{15}\) are unequal, ruling out an exponential curve. The curve is a hyperbola.
Hyperbola option is correct.
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