a^(x)=((a)/(k))^(y)=k^(m)" ,then "(1)/(x)-(1)/(y)

if a^(x)=((a)/(k))^(y)=k^(m) then prove that (1)/(x)-(1)/(y)=(1)/(m)

det [[x ^ (k), x ^ (k + 2), x ^ (k + 3) y ^ (k), y ^ (k + 2), y ^ (k + 3) z ^ (k) , z ^ (k + 2), z ^ (k + 3)]] = (xy) (yz) (zx) {(1) / (x) + (1) / (y) + (1) / ( from)}

The number of real values of k for which the lines (x)/(1)=(y-1)/(k)=(z)/(-1) and (x-k)/(2k)=(y-k)/(3k-1)=(z-2)/(k) are coplanar, is

The number of real values of k for which the lines (x)/(1)=(y-1)/(k)=(z)/(-1) and (x-k)/(2k)=(y-k)/(3k-1)=(z-2)/(k) are coplanar, is

The number of real values of k for which the lines (x)/(1)=(y-1)/(k)=(z)/(-1) and (x-k)/(2k)=(y-k)/(3k-1)=(z-2)/(k) are coplanar, is

Prove that if P(x,y) be any point on the line segment joining P_(1)(x_(1),y_(1)) and P_(2) (x_(2),y_(2)) then x=x_(1)+k(x_(2)-x_(1)) and y=y_(1)+k(y_(2)-y_(1)) , where bar (P_(1)P), bar (P_(1)P_(2))=k What conclusion can you draw about the position of P if 0 lt k lt 1

Prove that if P(x,y) be any point on the line segment joining P_(1)(x_(1),y_(1)) and P_(2) (x_(2),y_(2)) then x=x_(1)+k(x_(2)-x_(1)) and y=y_(1)+k(y_(2)-y_(1)) , where bar (P_(1)P), bar (P_(1)P_(2))=k What conclusion can you draw about the position of P if k gt 1

Prove that if P(x,y) be any point on the line segment joining P_(1)(x_(1),y_(1)) and P_(2) (x_(2),y_(2)) then x=x_(1)+k(x_(2)-x_(1)) and y=y_(1)+k(y_(2)-y_(1)) , where bar (P_(1)P), bar (P_(1)P_(2))=k What conclusion can you draw about the position of P if k=1

In X-Y plane, the path defined by the equation (1)/(x^(m))+(1)/(y^(m)) +(k)/((x+y)^(n)) =0 , is (a) a parabola if m = (1)/(2), k =- 1, n =0 (b) a hyperbola if m =1, k =- 1, n=0 (c) a pair of lines if m = k = n =1 (d) a pair of lines if m = k =- 1, n =1