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  • The middle point of the line segment joining (3, −1) and (1, 1) is shifted by two units (in the sense of increasing y) perpendicular to the line segment. Find the co-ordinates of the point in

The middle point of the line segment joining (3, −1) and (1, 1) is shifted by two units (in the sense of increasing y) perpendicular to the line segment. Find the co-ordinates of the point in the new position.

Option: 1

\mathrm{(4,\sqrt{2})}


Option: 2

\mathrm{(2+\sqrt{2},\sqrt{2})}


Option: 3

\mathrm{(2+\sqrt{2},2)}


Option: 4

\mathrm{(2-\sqrt{2},2)}


Answers (1)

best_answer

Let P be the middle point of the line segment joining

\mathrm{A(3,-2)\: and\: B(1,1)}

Then

\mathrm{P=\left(\frac{3+1}{2}, \frac{-1+1}{2}\right)=(2,0)}

Let P be shifted to Q where PQ = 2 and y co-ordinate of Q is greater than P          

Now slope of AB = –1

\therefore slope of PQ = 1

Co-ordinates of Q by distance formula \mathrm{=(2 \pm 2 \cos \theta, 0 \pm 2 \sin \theta) \text { where } \tan \theta=1}

                                                               =\left(2 \pm 2 \cdot \frac{1}{\sqrt{2}}, \quad 0 \pm 2 \cdot \frac{1}{\sqrt{2}}\right)=(2 \pm \sqrt{2}, \pm \sqrt{2})

as y co-ordinates of Q is greater than that of P.

Hence, \mathrm{Q=(2+\sqrt{2}, \sqrt{2})} is the required point.

Posted by

Rakesh

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