Question

for wht value k will be consecurive form ak+1,ak+2,ak-1 from an A.P

Answer 1

like that::-Well if these points are collinear it means that they lie on the same line.And no three points lying on the same straight line can form a triangle.In other words the area of such a triangle would be zero.Let’s say these 3 points are denoted byA(ax, ay), B(bx, by) and C(cx, cy)Thusax = -k+1ay = 2kbx = kby = 2–2kcx = -4-kcy = 6–2kLet the area of the triangle formed by these 3 points be A which is given byA = | ax(by-cy) + bx(cy-ay) + cx(ay-by)| x 0.5Now,by - cy = 2–2k-(6–2k)= 2–2k-6+2k = -4=>ax(by-cy)= (-k+1)(-4) =4k-4cy - ay = 6–2k-(2k) = 6–4k=>bx(cy-ay)= k(6–4k) =6k-4k.kay - by = 2k-(2–2k) = 2k-2+2k = 4k-2=>cx(ay-by)= (-4-k)(4k-2) = -16k+8–4k.k+2k= -14k+8–4k.kThus A = |4k-4+ 6k-4k.k-14k+8–4k.k | x 0.5 = 0 (area of triangle is zero)=> | -4k +4 - 8k.k | x 0.5 = 0=> -8k.k - 4k + 4 = 0=> -2k.k -k +1 = 0=> 2k^2 + k - 1 = 0=> 2k^2 + 2k - k - 1 = 0=> 2k(k+1) - 1(k+1) = 0=>(2k-1)(k+1) = 0Thus either 2k-1=0=> k = 1/2or k+1 = 0 => k = -1...hope help u☺☺