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[spikestar]

can some 1 plz help me with this practise question.
Question 4 part d(i & ii)

here is the link, http://www.vcaa.vic.edu.au/vce/studies/mathematics/specialist/pastexams/2004specmaths2.pdf

thanks for the help in adance

 

[icycloud]

Note: I'm using boldface to indicate vectors.

Part (i)

Firstly, we have OD = di + 0j and OQ = -i -2j
DQ = OQ - OD
= (-1-d)i - 2j

Part (ii)

Notice that DM is a diameter of the circle PMQD. This is because of the symmetry between P and Q, thus the x-axis cuts the circle into two semi-circles.

Thus, we have Q as a point on the circle, and since D and M both lie on the diameter, the angle subtended by DM at Q is pi/2.

We have the scalar product of DQ and MQ:

DQ (dot product) MQ = x₁x₂ + y₁y₂
= (-1-d)*(-5) + (-2)(-2)
= 5+5d +4
= 9+5d

However, since the included angle is pi/2, we use the scalar product theorem:

DQ (dot product) MQ = |DQ| * |MQ| cos (pi/2)
= 0

Equating the two, we have 9+5d = 0

And thus, d = -9/5 = -1.8