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Question

The area of the triangle formed by the positive x- axis and the normal and tangent to the circle x2+y2=4 at (1,3) is _________________.

A
23 sq.units
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B
32 sq.units
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C
6 sq.units
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D
none of these
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Solution

The correct option is A 23 sq.units

The given equation of circle is x2+y2=4
The equation of the normal to the circle at (1,3) is same as the line joining the points (1,3) and (0,0), which is given by

y3x1=3010

y3x1=3

y3=3x3

y=3x

So, the slope of normal is 3.
We know that the product of the slope of the tangent and normal is 1.
The slope of tangent is 13
Now, the equation of the tangent to the circle at (1,3) is given by
y3x1=13
3y3=x+1
y=(x+4)3 ----- ( 2 )
Putting y=0 in ( 2 ), we get x=4.
Thus, ABC is triangle formed by the positive xaxis and tangent and normal to the given circle at (1,3)
Now,
Area of AOB= Area of AOM+ Area of AMB
=10ydx+41ydx

=103xdx+41(x+43)dx

=[3x22]10+41x3dx+4143dx

=(320)[x223]41+[43x]41
=321623+123+16343
=32+332
=23



1489835_202333_ans_da1bbf3d01b7404d95e4fb58875314b5.jpeg

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