C1 Circles - I need some help please

Hi There,

I really need some help with questions 13, 14 and 15 in the OCR C1&C2 textbook. It is on page 207. I really need some help as I have been struggling all day with this question.

Reply 1

NotNotBatman

Original post by phat-chewbacca

Hi There,

I really need some help with questions 13, 14 and 15 in the OCR C1&C2 textbook. It is on page 207. I really need some help as I have been struggling all day with this question.

What are the questions, could you write them or post a picture?

Reply 2

the bear

if you post the questions we can help you ?

Reply 3

phat-chewbacca

Here you go. Thanks a lot.

IMG_0472.jpg524.8KB

Reply 4

Ed5

13) You need to complete the square to factorise it into a circle equation

(x - a)^2 + (y - b)^2 = r^2

where (a,b) are the center coordinates and r is the radius.

x^2 + y^2 - 2rx - 2ry + r^2 = (x - r)^2 - r^2 + (y - r)^2 - r^2 + r^2 = 0 \Rightarrow (x-r)^2 + (y-r)^2 = r^2

Here, we've proved the circle's center is (r,r) and its radius is r, hence it is touching both axis!

14) Complete the square again:

(x-7)^2 - 49 + (y-5)^2 - 25 + c = 0 \Rightarrow (x-7)^2 + (y-5)^2 = 74 - c

From this, we know the circle is centered at (7,5). Hence, in order to lie wholly in the first quadrant (AKA the top right quarter), its radius can't be any larger than 5.

r = \sqrt{74-c} \leq 5 \Rightarrow 74 - c \leq 25 \Rightarrow 49 \leq c

That gives you the lower bound of c. To get the higher bound, take a look at the equation for the radius squared, 74 - c. If c was at or above 74, the radius would become zero or negative, hence this is our upper bound.

15 will be similar to the above two: have a go and see if you can solve it.

(edited 7 years ago)

Reply 5

NotNotBatman

Original post by phat-chewbacca

Here you go. Thanks a lot.

For 13.) Complete the square and write it in the form

(x-a)^2 + (y-b)^2 =r^2

, then set x and y equal to 0. the centre coordinates should be the same as the radius' length.

14.) consider the center and how long the radius could be

15.) Again consider the center and the radii of each circle and show they lie outside of each other.

Reply 6

phat-chewbacca

Original post by NotNotBatman

For 13.) Complete the square and write it in the form

(x-a)^2 + (y-b)^2 =r^2

Again I am completed stumped. For every question I have always managed to complete the square and find the centre and radius, but I just don't understand the last part to these questions. I just don't understand the concept of considering the length of the radius. arghhh

Reply 7

NotNotBatman

Original post by phat-chewbacca

If a circle touches both axes, it touches them at the same distance from another point. The distance from the centre to the points where the circle touches the axes are the same, this is also the radius.

@Ed5 Has given a good explanation above!

Reply 8

RDKGames

Study Forum Helper

Original post by phat-chewbacca

Here you go. Thanks a lot.

Question's were well explained by user's above, however for Q15 I want to add that for two circles to lie OUTSIDE each other, the sum of their radii must be less than or equal to the distance between the two centres.

Consider circle 1 with centre (a,b) and radius r:

(x-a)^2+(y-b)^2=r^2

and circle 2 with centre (c,d) with radius R:

(x-c)^2+(y-d)^2=R^2

Then circle 1 and 2 lie outside each other if, and only if:

r+R \leq \sqrt{(a-c)^2+(b-d)^2}

Can you visualise this? Think of 2 arbitrary circles with a line joining their two centres. When these two circles are tangent, sum of radii equal the distance between the centres. Move one closer to the other, and then the sum is greater than the distance. Move them away and then the sum is less than the distance.

(edited 7 years ago)

Reply 9

phat-chewbacca

Original post by RDKGames

(x-a)^2+(y-b)^2=r^2

and circle 2 with centre (c,d) with radius R:

(x-c)^2+(y-d)^2=R^2

Then circle 1 and 2 lie outside each other if, and only if:

r+R \leq \sqrt{(a-c)^2+(b-d)^2}

I am finding it really hard to visualise this. Do you know of any videos? Also if you don't mind could you think of all the things that I need to know for C1 Circles, and post them on here.

Reply 10

RDKGames

Study Forum Helper

Original post by phat-chewbacca

I am finding it really hard to visualise this. Do you know of any videos? Also if you don't mind could you think of all the things that I need to know for C1 Circles, and post them on here.

No I do not know of any videos. It just came to me when I looked at the question and just quickly derived the conditions in a general case.

You can take a look at this I made for you. You can move the circles around by pressing on their centres and moving them. The panel on the left gives you the information you need.

D

is the distance between the centres and

S

is the sum of their radii. Try moving one circle closer and closer to the other and observe what happens between the two numbers.

https://www.desmos.com/calculator/6owhiberoy

Now the only thing here you may wish to imagine is that a portion of the blue line is contais the radii for both circles, from their centres to the point here the line cuts the circle; ie the circumference. I would've displayed this for you but it gets too hairy on desmos with generality.