# 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)#?

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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)# ?

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

From the figure solve for x:

Rearranging and solving for x we have:

Applying the same logic to the ratio:

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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} ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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- Express the area of a triangle given by vertices, #A(x_1,y_1), B(x_2,y_2), C(x_3,y_3)#. Show that it can be expressed as determinant of: #det(Delta) = [(1, 1, 1 ),(x_1, x_2, x_3),(y_1, y_2, y_3) ]#.Calculate the area of A(3,6), B(7,8), & C(5,2)?
- Circle A has a center at #(1 ,7 )# and a radius of #3 #. Circle B has a center at #(-3 ,-2 )# and a radius of #2 #. Do the circles overlap? If not, what is the smallest distance between them?
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- Circle A has a center at #(1 ,4 )# and an area of #100 pi#. Circle B has a center at #(7 ,9 )# and an area of #36 pi#. Do the circles overlap? If not, what is the shortest distance between them?

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