Question

please help me , I can not solve these problems given in the photo which are Q no.11,12,13,14,15&16.

Answer 1

11.  given    a = half of  4
12.  AB is the line joining (0,5), (5,0)..  x + y = 5
      So (x,y) integers should be such that 0< x+y < 5.
      (1,1), (1,2), (1,3), (2,2), (3,1) 
13.  If you know about slopes of straight lines, then it is easy.  Otherwise, try solving it using distances.  But that is longer.

       slope of PQ = 6-2)/(4-1) = 4/3 = slope of RS = (b-7)/(a-5)
        4a = 3b - 1
       slope of QR = (7-6)/(5-4) = 1 = slope of SP = (b-2)/(a-1)
         a  = b - 1
         solve to get:   a = 2, b = 3
       
      distance : PQ² = 3²+4² = 25 = RS² = (b-7)²+(a-5)²     ---(1)
       distance  QR² = 1 +1 =  2   = SP² = (b-2)²+(a-1)²     --- (2)
          (2) - (1) gives:   - 23 = 10 b -45 + 8 a - 24
                                  4 a + 5 b = 23
            only option (c) fits this equation.

14. shortest distance of (2,4) from x axis is  y coordinate 4.  so ans = 0.

15.  y coordinate  | k |   the point could be on either side. distance is +ve.

16.  orthocenter is pt of intersection of altitudes. (0,0) is the point.
         as  x axis and y axis are altitudes of right angle triangle.