Question

# Let y=y(x) be a curve passing through the point (1,1) and satisfying dydx+√(x2−1)(y2−1)xy=0. If the curve passes through the point (√2,k), then the largest value of |[k]| is (Here, [.] represents the greatest integer function)

A
2
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B
1
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C
5
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D
7
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Solution

## The correct option is A 2dydx+√(x2−1)(y2−1)xy=0 ⇒y√y2−1dy=−√x2−1xdx ⇒∫y√y2−1dy=−∫x2−1x√x2−1dx Let y2−1=t2⇒2ydy=2tdt ∴∫dt=−∫x√x2−1dx+∫1x√x2−1dx ⇒t=−√x2−1+sec−1x+C ⇒√y2−1=−√x2−1+sec−1x+C The curve passes through (1,1). ∴C=0 Hence, the curve is √y2−1=−√x2−1+sec−1x Also, (√2,k) lies on the curve. ⇒√k2−1=−1+π4 ⇒k2=(π4−1)2+1 ⇒k2=m+1 for some m∈(0,1) ⇒k=±√m+1 ⇒[k]=1 or −2 ∴ The largest value of |[k]| is 2

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