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Question

Statement I: The point A(3,1,6) is the mirror image of the point P(1,3,4) in the plane x-y+z=5.

Statement II: The plane x-y+z=5 bisects the line segment joining A(3,1,6) and B(1,3,4) .


A

Statement I is correct, Statement II is correct; Statement II is the correct explanation for Statement I

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B

Statement I is correct, Statement II is correct; Statement II is not the correct explanation for Statement I

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C

Statement I is correct, Statement II is incorrect

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D

Statement I is incorrect, Statement II is correct

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Solution

The correct option is B

Statement I is correct, Statement II is correct; Statement II is not the correct explanation for Statement I


Explanation of correct option.

Finding the value of a,b, and c and finding the midpoint of the points given in statement II::

We can find the image of any point using the image formula. Hence, using the image formula for the point p,q,r=(1,3,4) and the plane lx+my+nz-s=0⇒x-y+z-5=0

The image formula is a-pl=b-qm=c-rn=-2lp+qm+rn+sl2+m2+n2

⇒a-11=b-3-1=c-41=-21-3+4-53⇒a-11=b-3-1=c-41=2∴(a,b,c)=(3,1,6)

Hence, Statement I is true.

Now,

The midpoint of the given points x1,y1,z1=A(3,1,6) and x2,y2,z2=B(1,3,4)

The midpoint formula is M=x1+x22,y1+y22,z1+z22

3+12,1+32,4+42=2,2,5

Substituting the midpoint in x-y+z=5

x-y+z=5⇒2-2+5=5⇒5=5

Hence, Statement II is true.

Therefore, the correct answer is option (B).


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