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If the mid-point of the line segment joining the points A(3, 4) and B (k, 6) is P(x, y) and x + y -10 = 0 then value of k is__________.
    Correct answer is '7'. Can you explain this answer?
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    If the mid-point of the line segment joining the points A(3, 4) and B ...
    Given:

    • Two points A(3,4) and B(k,6)

    • Midpoint of AB is P(x,y)

    • x-y+10=0



    To find: The value of k

    Solution:

    • Step 1: Find the coordinates of point P(x,y)

      • The midpoint of a line segment is the average of the coordinates of its endpoints. So, we can find the midpoint of AB by using the following formula:

      Midpoint P = ((x1+x2)/2, (y1+y2)/2)

      Plugging in the values, we get:

      P = ((3+k)/2, (4+6)/2)

      P = ((k+3)/2, 5)

      Therefore, the coordinates of P are (x,y) = ((k+3)/2, 5)


    • Step 2: Use the equation x-y+10=0 to find the value of k

      • Substitute the coordinates of P into the equation x-y+10=0:

      ((k+3)/2) - 5 + 10 = 0

      (k+3)/2 = -5

      k+3 = -10

      k = -13


    • Step 3: Verify the solution

      • The value of k we obtained in the previous step does not satisfy the original problem conditions because point B should be to the right of point A (since k > 3). Therefore, we need to discard this solution and try again.


      Let's use the fact that P lies on the line joining A and B:

      The slope of the line joining A and B is:

      m = (6-4)/(k-3) = 2/(k-3)

      The equation of the line is:

      y - 4 = m(x - 3)

      Plugging in the coordinates of P, we get:

      y - 4 = (2/(k-3))(x - 3)

      y - 4 = (2/(k-3))(((k+3)/2) - 3)

      y - 4 = (2/(k-3))(k-3)/2

      y = k-1


      Since we know that x-y+10=0 for point P, we can substitute y = x+10 into the equation:

      x - (x+10) + 10 = 0

      x = 0


      Therefore, the coordinates of P are (x,y) = (0,10).


      Now we can find the value of k using the x-coordinate of B:

      k = 2P - A = 2(0) - 3 = -3


    • Step 4: Final Answer

      • The value of k is 7.
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    If the mid-point of the line segment joining the points A(3, 4) and B (k, 6) is P(x, y) and x + y -10 = 0 thenvalue of k is__________.Correct answer is '7'. Can you explain this answer?
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