a)A circle has centre C(-1, 3) and
radius 5 units.
(a) Find the equation of the circle
(b) The line 3x — y + 1 = O 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 angleZXY = 90 degrees.
Please help
Hahah, thanks for clearing up the typo. I was getting so confuseda)
Equation: (x-h)^2 + (y-k)^2 = r^2
so it is (x+1)^2 + (y-3)^2 = 25
b)
Use simultaneous equations by substitution
3x - y + 1 = 0
y = 3x + 1, sub this into the circle
(x+1)^2 + (3x-2)^2 = 25
x^2 + 2x + 1 + 9x^2 - 12x + 4 - 25 = 0
10x^2 - 10x - 20 = 0
x^2 - x - 2 = 0
(x-2)(x+1) = 0
x = 2, -1
sub these into the line equation to get y = 7, -2
so the answer is (2, 7) and (-1, -2)
c) Don't understand the question
Edit: typo
Do you have a diagram for part c so I can solve it? ThanksHahah, thanks for clearing up the typo. I was getting so confused
Thank you so much for helping ^_^
c)A circle has centre C(-1, 3) and
radius 5 units.
(a) Find the equation of the circle
(b) The line 3x — y + 1 = O 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 angleZXY = 90 degrees.
Please help
I am basing my answers off of Pikachu's for Part a and b(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 angleZXY = 90 degrees.
Please help
Lol yeah, you can find the gradients like that. I always forget and find the equation and then the gradient...c)
X(2,7) and Y(-1,-2) and Z(x,y)
So we know YZ passes through the diameter, ie YC = CZ so C is the midpoint of YZ
Using midpoint formula,
(-1 = (x-1)/2, 3 = (y-2)/2)
= (x = -1, y = 8)
Therefore Z(-1, 8)
d) Find gradients of line ZX and XY
m of ZX = (8-7)/(-1-2)
= -1/3
m of XY = (-2-7)(-1-2)
=
3
m1 x m2 = -1/3 x 3
= -1 hence they're perpendicular so angle ZXY = 90 degrees
Awesome. Once again, thank you.c)
X(2,7) and Y(-1,-2) and Z(x,y)
So we know YZ passes through the diameter, ie YC = CZ so C is the midpoint of YZ
Using midpoint formula,
(-1 = (x-1)/2, 3 = (y-2)/2)
= (x = -1, y = 8)
Therefore Z(-1, 8)
d) Find gradients of line ZX and XY
m of ZX = (8-7)/(-1-2)
= -1/3
m of XY = (-2-7)(-1-2)
=
3
m1 x m2 = -1/3 x 3
= -1 hence they're perpendicular so angle ZXY = 90 degrees
Oh, cool. More explanations! Thank you!I am basing my answers off of Pikachu's for Part a and b
c) Y must be (-1,-2) (directly below c(-1-3))
If YZ are a diameter, Z is directly 10 units above Y, since the centre is between them
Therefore, Z(-1,8)
d) For ZXY to be 90 degrees, ZX must be perp. to XY
Eqn of XY is -y=-3x-1 (m=3)
Eqn of XZ is -3y=x-23 (m=-1/3)
If both gradients multiply to equal -1, then the lines are perpendicular
-1/3*3=-1
Therefore, they are perp. and ZXY is right angle (90 degrees)
There is a shortcut to this question:
Since ZY is a diameter and X is a point that lies on the circumference of the circle, ZXY has to be a right angle
But this is a circle theorem and you learn it in preliminary extension 1 (I assume these questions are just Preliminary Mathematics and they would rather you use the first method)
Hope this helps and that I got them right lol
3U book encompasses 2u, you don't need the 2u book.I have the 2 u and 3 u Cambridge textbooks, and i dont know which one i should do. Should i go straight to the 3u book or do 2u and then 3u.
2u: Geometric applications of calculusBTW to all 2017ers, what topic(s) do you start when you come back to school?
Didn't you already do curve sketching in calculus?2u: Geometric applications of calculus
3u: harder parametrics and circle geo/ext 1 curve sketching
4u: complex numbers
3u: Probably the miscellaneous stuff like inequalities, angles between lines, or circle geoBTW to all 2017ers, what topic(s) do you start when you come back to school?
Is this after your HSC exams?3u: Probably the miscellaneous stuff like inequalities, angles between lines, or circle geo
4u: Complex numbers probably
Yep, 3u and 4u don't start till week 3 because the HSC for maths is week 2 Friday.Is this after your HSC exams?