If #P(x,y)# lies on the interval #A(x_1,y_1), B(x_2,y_2)# such that #AP : PB =a : b#, with a and b positive,show that
#x= (bx_1+ax_2) /(b+a)#
and #y=(by_1+ay_2)/(b+a)#?
.

Question

If #P(x,y)# lies on the interval #A(x_1,y_1), B(x_2,y_2)# such that #AP : PB =a : b#, with a and b positive,show that
#x= (bx_1+ax_2) /(b+a)#
and #y=(by_1+ay_2)/(b+a)#?
.

Sadie Allen · Accepted Answer

See below.

If #P in bar(AB)# then #P =A +lambda (B-A), lambda in [0,1]# or equivalently #P = B + mu(A-B), mu in [0,1]# also #norm(P-A) = lambda norm(B-A)# and #norm(P-B) = mu norm(A-B)# but #norm(P-A)/norm(P-B) = lambda/mu = a/b = (a(a+b))/(b(a+b))# but anyway #lambda = a/(a + b)# and #1-lambda = b/(a+b)# so another reading is #P = (1-lambda)A+lambda B=b/(a+b)A+a/(a+b)B# resulting in #x = (b x_1+a x_2)/(a+b)# #y = (b y_1+a y_2)/(a+b)#

Claire Akin · Answer

See the geometric construction to guide you with set up...

From the figure solve for x:

#ax_2-ax_1=(a+b)(x-x_1)=ax-ax_1+bx-x_1=(a+b)x-(a+b)x_1 #
Rearranging and solving for x we have:
#x= (ax_2+bx_1)/(a+b)#

Applying the same logic to the ratio:
#a/(a+b) = bar(PC)/bar(BD)= (y-y_1)/(y_2-y_1)# Solving for #y#:
#y = (ay_2+by_1)/(a+b)#

Noah Atkinson · Answer

undefined Given point $ P(x, y) $ lies on the interval $ AB $ such that $ AP : PB = a : b $, where $ a $ and $ b $ are positive, we need to show that:

$$ x = \frac{bx_1 + ax_2}{b + a} $$
$$ y = \frac{by_1 + ay_2}{b + a} $$

Let's start by finding the coordinates of point $ P $. Using the section formula, we have:

$$ x = \frac{bx_1 + ax_2}{a + b} $$
$$ y = \frac{by_1 + ay_2}{a + b} $$

Now, let's verify if $ x = \frac{bx_1 + ax_2}{b + a} $ and $ y = \frac{by_1 + ay_2}{b + a} $.

$$ \frac{bx_1 + ax_2}{b + a} = \frac{a(bx_1 + ax_2)}{a(b + a)} + \frac{b(bx_1 + ax_2)}{b(b + a)} $$
$$ = \frac{abx_1 + a^2x_2 + b^2x_1 + abx_2}{a^2 + ab + ab + b^2} $$
$$ = \frac{(a + b)(bx_1 + ax_2)}{(a + b)^2} = \frac{bx_1 + ax_2}{a + b} $$

Similarly for $ y $, we get:

$$ \frac{by_1 + ay_2}{b + a} = \frac{ay_1 + by_2}{a + b} $$

Hence, we've shown that $ x = \frac{bx_1 + ax_2}{b + a} $ and $ y = \frac{by_1 + ay_2}{b + a} $.

If #P(x,y)# lies on the interval #A(x_1,y_1), B(x_2,y_2)# such that #AP : PB =a : b#, with a and b positive, show that #x= (bx_1+ax_2) /(b+a)# and #y=(by_1+ay_2)/(b+a)#?

If #P(x,y)# lies on the interval #A(x_1,y_1), B(x_2,y_2)# such that #AP : PB =a : b#, with a and b positive,show that #x= (bx_1+ax_2) /(b+a)# and #y=(by_1+ay_2)/(b+a)#? .

If #P(x,y)# lies on the interval #A(x_1,y_1), B(x_2,y_2)# such that #AP : PB =a : b#, with a and b positive,show that
#x= (bx_1+ax_2) /(b+a)#
and #y=(by_1+ay_2)/(b+a)#?
.