Prelim. 3U two math questions (1 Viewer)

[Future-Med-GP]

Could anyone please assist me and post your working out and solutions to the following questions? It's from a 2U/3U book, so it should be simple, but I'm stuck ... Thanks.

1) A, B, C are collinear points AB = BC. If A is the point (4,5) and B is the point (1, -1), find the coordinates of C.

2) Given A(2,3) and B(7,-4), find the coordinates of P such that:

AP/PB = 2/5

 

[vds700]

Could anyone please assist me and post your working out and solutions to the following questions? It's from a 2U/3U book, so it should be simple, but I'm stuck ... Thanks.

1) A, B, C are collinear points AB = BC. If A is the point (4,5) and B is the point (1, -1), find the coordinates of C.

2) Given A(2,3) and B(7,-4), find the coordinates of P such that:

AP/PB = 2/5

1. Let C be (a, b). Use division of an interval into a given ratio formula. B divides AC in the ratio 1:1.

x = (nx₁ + mx₂)/(m + n)
1 = (a + 4)/2
a + 4 = 2
a = -2

y = (ny₁ + my₂)/(m + n)
-1 = (b + 5)/2
b + 5 = -2
b = -7.

Therefore C is (-2, -7)

2. Im not sure about this one. Are u sure they didn't say in the question to find the locus of P. I end up getting the locus as an ellipse which is 4 unit level maths. Maybe theres a simpler way of doing it.

 

[Iruka]

q2) should give you a circle, I think. Check your working out.

 

[vds700]

q2) should give you a circle, I think. Check your working out.

yeah you're right

 

[lolokay]

2) Given A(2,3) and B(7,-4), find the coordinates of P such that:

AP/PB = 2/5

P is a collinear point, not a locus btw

It either divides the line AB internally or externally in the ratio of 2 to 5,

if internally, then point P is made up of 2/7 of point B and 5/7 of point A (it approaches B from A and gets 2/7 of the way there, therefore with 5/7 of the way to go and thus AP/PB = 2/5) or think of it as P = A + 2/7(B - A)

2/7 * 7 + 5/7 * 2 = 24/7 , and 2/7 * -4 + 5/7 * 3 = 1
so one such P is (24/7, 1)

if it divides AB externally, then it is made up of -2/3 of B and 5/3 of A (moves from A away from B until it is 2/3 of the distance AB away from A - and therefore 5/3 of AB away from B, so AP/PB = 2/5), or P = A + -2/3(B - A)

-2/3 * 7 + 5/3 * 2 = -4/3 , and -2/3 * -4 + 5/3 * 3 = 23/3
so the other such P is (-4/3, 23/3)