Help me! Urgent! Thanks?

Question

With respect to the origin O, the position vectors of points A, B and C are (2, 3), (4, -2) and (-3, -3) respectively.

(i) A point E (-2, p) is such that 2 |CE| = |AB|. Find the possible values of p.
Ans: - 5 1/2 or -1/2

(ii) Find the coordinates of the point Q if 2 CQ = BA + 1/2 AC.
Ans: (-5 1/4, -2)

Please help by showing workings...Any help will be greatly appreciated!

Answer 1

If V is a vector with coordinates (x,y), then |V| is
|v| = sqrt(x^2 + b^2)

Looking at (i):

CE is the vector from C to E or (-2, p) - (-3, -3) = (1, p + 3)
AB is the vector from A to B or (4, -2) - (2, 3) = (2, -5)

2|CE| = |AB| means
2sqrt(1^2 + (p+3)^2) = sqrt(2^2 + (-5)^2)

square both sides to get rid of the square roots and get:

4(1 + (p+3)^2) = (4 + 25)

This is a quadratic in p and so can be solved for p.

Now for part (ii)

Let Q be the have the coordinates (x,y), then

2 CQ = 2((x,y) - (-3, -3)) = 2 (3+x, 3+y)
BA = (4, -2) - (2, 3) = (2, -5)
AC = (-3, -3) - (2, 3) = (-5, -6)

BA + (1/2)AC = (2, -5) + (-5/2, -3) = (-1/2, -8)

so 2 (3+x, 3+y) = (-1/2, -8) or
2(3+x) = -1/2
2(3 + y) = -8

My answers don't match yours, so either my arithemetic is off or your answers are. But the basic idea is right.