(lambda)/(1)=-((ax_(1)+by_(1)+c))/((ax_(2)+by_(2)+c))

The points P divides the line - segment joining the points A(x_(1),y_(1))andB(x_(2),y_(2)) in the ratio m : n . If P lies on the line ax+by +c=0 , prove that, (m)/(n)=-(ax_(1)+by_(1)+c)/(ax_(2)+by_(2)+c) .

A : The image of the origin with respect to the line x+y+1=0 is (-1, -1) R : If (h, k) is the image of (x_(1), y_(1)) with respect to the line ax+by+c=0 then (h-x_(1))/(a)=(h-k_(1))/(b)=(-2(ax_(1)+by_(1)+c))/(a^(2)+b^(2))

A : The foot of the perpendicular from (3, 4) on the line 3x-4y+5=0 is (81//25, 92//25) R : If (h, k) is the foot of the perpendicular from (x_(1), y_(1)) to the line ax+by+c=0 then (h-x_(1))/(a)=(h-k_(1))/(b)=(-(ax_(1)+by_(1)+c))/(a^(2)+b^(2))

A : The ratio in which the perpendicular through (4, 1) divides the line joining (2, -1), (6, 5) is 5:8 . R : The ratio in which the line ax+by+c=0 divides the line segment joining (x_(1), y_(1)), (x_(2), y_(2)) is (ax_(1)+by_(1)+c): -(ax_(2)+by_(2)+c) .

If Q(h,k) is the image of the point P(x_(1),y_(1)) w.r.to the straight line ax+by+c=0 then prove that (h-x_(1)):a=(k-y_(1)):b=-2(ax_(1)+by_(1)+c):a^(2)+b^(2) or (h-x_(1))/(a)=(k-y_(1))/(b)=(-2(ax_(1)+by_(1)+c))/(a^(2)+b^(2)) and find the image of (1,-2) w.r.t the straight line 2x-3y+5=0

If (h,k) is the foot of the perpendicular from (x_(1),y_(1)) to the line ax+by+c=0, (a!=0,b!=0) then (h-x_(1))/(a)=(k-y_(1))/(b)=(-(ax_(1)+by_(1)+c))/(a^(2)+b^(2)) Find the foot of the perpendicular from (-1,3) to the line 5x-y=18

Show that the plane ax+by+cz+d=0 divides the line joining the points (x_(1),y_(1),z_(1)) and (x_(2),y_(2),z_(2)) in the ratio (ax_(1)+by_(1)+cz_(1)+d)/(ax_(2)+by_(2)+cz_(2)+d)

If Q(h,k) is the foot of the perpendicular of P(x_(1),y_(1)) on the line ax+by+c=0 then prove that (h-x_(1)),a=(k-y_(1)),b=-(ax_(1)+by_(1)+c):(a^(2)+b^(2)) .