Circle C: Centre, Radius & Point D

In summary, the circle with equation x^2+y^2+6x-16=0 has a centre of (-3,0) and a radius of 5. Point A, (0,4) lies on the circle. To find the coordinates of point D, which lies diametrically opposite to point A, a vector from the center to A is created and then doubled to get a vector from the center to D. Using this vector, the coordinates of D are found to be (-6,-4). It is also possible to use congruent triangles to find the coordinates of D. The equation of the circle can also be expressed as (x + 3)^2 + y^2 = 25, with x
  • #1
david18
49
0
Circle C has equation [itex] x^2+y^2+6x-16=0 [/itex]

i) find centre and radius (turned out to be centre (-3,0) and radius 5)
ii)verify that point A, (0,4) lies on C
iii)Find the coords of point D, given that AD is a diameter of C

I can do parts i) and ii) and for part iii) would I just use a simultaneous equation with the equation of the line from centre to point A (which worked out as 4/3x+4) and the equation of the circle? Because it seems a bit long winded
 
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  • #2
Point D is at the other end of the circle, with respect to A. How would you express its coordinates?
 
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  • #3
I can work it out easily if i sketch the circl;e and it comes out as (-6,-4), but there must be some kind of method to obtain the answer without the sketch.
 
  • #4
Let me put it this way. Imagine a vector from the center to A. What is the vector from the center to D?
 
  • #5
You can subtract two points to obtain a vector:

Make a vector [tex]\vec{OA}[/tex] by doing A-O, where O=(0,0), then actually [tex]\vec{OA}[/tex] becomes (0,4).

Then let S be the centre of the circle, so S = (-3,0)
And make a vector pointing from A to S [tex]\vec{AS}[/tex] = S - A = (-3,-4)

Then just use add vectors like this:
[tex]\vec{OA}[/tex] + [tex]\vec{AD}[/tex] = [tex]\vec{OD}[/tex]
but because [tex]\vec{AD}[/tex] is diameter it means that:
[tex]\vec{AD}[/tex] = 2*[tex]\vec{AS}[/tex]

So it becomes:
[tex]\vec{OA}[/tex] + 2*[tex]\vec{AS}[/tex] = [tex]\vec{OD}[/tex]
From there you have [tex]\vec{OD}[/tex] = (-6,-4)
And because [tex]\vec{OD}[/tex] = D - O it is the point D, and its coordinates are (-6,-4)
 
  • #6
You could use congruent triangles. If A= ([itex]x_1[/itex],[itex]y_1[/itex]) is on the circle with center O= ([itex]x_0[/itex],[itex]y_0[/itex]), and AB is a diameter, with B= ([itex]x_2[/itex],[itex]y_0[/itex], then The triangle with vertices A, O, and ([itex]x_0[/itex],[itex]y_1[/itex]) is congruent to the triangle with vertices B, O, and ([itex]x_0[/itex], [itex]y_1[/itex]).
 
  • #7
From the centre and radius you have worked out, express the equation of the circle in co-ordinates (x’, y) (you are lucky you have to change only one co-ordinate, the centre lies on the x axis) that have the centre as origin. From your values you know it has got to be (x + 3)^2 + y^2 = 25. Check that this is the same as your original equation. Call it
x’^2 + y^2 = 25

where x’ = (x + 3).

What is the point diametrically opposite to a point A coordinates (x’, y) on a circle centred on origin? Easy -> (-x’, -y)

Now you have that in the new co-ordinates, what is it expressed in the old ones? See last equation.
 

1. What is the center of a circle?

The center of a circle, denoted as C, is the point in the middle of the circle where all points on the circle are equidistant from. It is also the point from which the radius of the circle is measured.

2. What is the radius of a circle?

The radius of a circle, denoted as r, is the distance from the center of the circle to any point on the circle. It is half of the diameter of the circle and is used to calculate the circumference and area of the circle.

3. How is the radius of a circle measured?

The radius of a circle can be measured using a ruler or any other measuring tool. Place the center of the ruler at the center of the circle and measure the distance to any point on the circle. This will give you the radius of the circle.

4. What is point D in relation to a circle?

Point D is simply any point on the circle. It is used in some equations and formulas to represent any point on the circumference of the circle.

5. How are the center, radius, and point D related in a circle?

The center of a circle is the point from which the radius is measured, and the circumference of the circle is the distance around the circle from any point on the circumference, such as point D. The radius is also used in equations and formulas to calculate the circumference and area of a circle.

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