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Question

If one end of the diameter is 1,1 and the other end lies on the line x+y=3, then the locus of the center of a circle is


A

x+y=1

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B

2xy=5

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C

2x+2y=5

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D

None of these

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Solution

The correct option is C

2x+2y=5


Explanation for the correct option.

Step 1: Given information

Here one end of the diameter is given and it is given that other end is on the line with equation x+y=3.

Now, let the x-coordinate of the other end be t then its y-coordinate will be 3-t. So, the coordinates of the other end will be t,3-t.

As the coordinates of both the ends of the diameter are known then the equation of the variable circle can be written as,

x-1x-t+y-1y-3-t=0x2-xt-x+t+y2-3y+yt-y+3-t=0x2+y2-1+tx-4-ty+3=0

We know that the general form of equation of circle is x2+y2+2gx+2fy+c=0, where -g,-f are the coordinates of the center of a circle.

After the comparison of general form of equation of circle and equation of variable circle we have,

2g=1+t12f=4-t2

Step 2 : Add the equations (1)and(2)

2g+2f=5

Therefore, the locus of the center of a circle is 2x+2y=5.

Hence, the correct option is (C).


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