Locus and parabola question (1 Viewer)

[MzG1zi]

i was looking over my prelim textbook and i discovered that i never did this question. After trying to solve it for half an hour, i thought i would just post it here.

the question is: A circle has centre C (-1, 3) and radius 5 units.
a) find the equation of the circle
b) the line 3x-y+1=0 meets the circle at two points. Find their coordinates.
c) Let the coordinates be X and Y, where Y is the coordinate directly below the centre C. Find the coordinates of point Z, where YZ is a diameter of the circle.
d) Hence show that angle ZXY= 90°

i did part a) and b) and got the same answers as in the back, but to solve c) i drew the circle and in my sketch, the two points where the line meets the circle [(2, 7) and (-1, -2)] , arent on the circle :/
...so im immensely confused
any help would be greatly appreciated

 

[Drongoski]

If you do it again you'd probably find you've made a silly mistake somewhere.

Using a typical grid-paper (from your usual school maths exercise book) using a compass & set square or ruler, construct the circle and the straight line (y = 3x + 1) you will find that X(2,7) and Y(-1,-2) rest nicely on the circle as they should.

The co-ords of point Z(p,q) can be easily found, noting C is the mid-pt of YZ, via:

(p-1)/2 = -1 [or since Y is directly below C, p is -1] and (q-2)/2 = 3

So Z is (-1,8)

So you have gradients of XZ = - 1/3 and of XY = 3 and (- 1/3) x 3 = -1 ) so
XZ and XY are perpendicular so that angle ZXY = 90 degrees (how do you get the degree symbol ??!!)

Hope this helps.

 

[MzG1zi]

Thanks heaps
as you said, a silly mistake. i made (3, -1) as the centre instead of the other way around...and i was wondering why (-1, -2) wasnt directly below C as i had previously calculated XD

just wondering, would it also be valid if i said angle ZXY= 90° because it is the angle in a semi-circle, or do i have to do the whole gradient(zx) x gradient(yx) = -1 thing?... not that i have anything against your method, just curious.

lol, nd for the degree symbol i just copied and pasted it off google... idk how to get it the proper way either.

 

[Drongoski]

Of course it is well-known that the angle in a semi-circle is 90 deg.

But in this case I think the author wants you to show this to be the case the way I've indicated.