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Question

At which of the following point/s is/are the tangent parallel to the y-axis on the curve x2+y2−2x−4y+1=0 ?

A
(-1, 2)
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B
(3, 2)
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C
(1, 4)
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D
(3, -2)
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Solution

The correct option is B (3, 2)
Let P(a, b) be the required point. As P(a, b) lies on the curve x2+y22x4y+1=0.a2+b22a4b+1=0 .....(1]If the tangent to the given curve at P is parallel to yaxis, then(dxdy)P=0The equation of the curve is x2+y22x4y+1=0.Differentiating both sides w.r.t. y, we get2x(dxdy)+2y2(dxdy)4=02dxdy(x1)=2(2y)dxdy=(2y)(x1)As (dxdy)P=0(2b)(a1)=0b=2Putting b=2 in (1] we geta2+42a8+1=0a22a3=0(a+1)(a3)=0a=1, 3Hence, the required points are (1, 2) and (3, 2).

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