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Question

Assertion :A normal is drawn at a point P(x,y) of a curve. It meets the x-axis and the y-axis in point A and B, respectively, such that 1OA+1OB=1, where O is the origin. The equation of such a curve passing through (5,4) is (x−1)2+(y−1)2=25 Reason: OA=x+ydydx and OB=x+ydydxdydx

A
Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
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B
Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
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C
Assertion is correct but Reason is incorrect
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D
Both Assertion and Reason are incorrect
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Solution

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
The equation of the normal at (x,y) is,
(Xx)+(Yy)dydx=0
x.ey(cosysiny)=eysiny+c
Xx+ydydx+Y(x+ydydx)dydx=1
OA=x+ydydx,OB=x+ydydxdydx
Also, 1OA+1OB=11+dydx=x+ydydx(y1)dydx+(x1)=0
Integrating, we get
(y1)2+(x1)2=c
Since the curve passes through (5,4),c=25
Hence, the curve is (x1)2+(y1)2=25

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