The line 3x + 4y = 24 cuts the x axis at point A and the y axis at point B. The point C is the mid-point of AB.

(i) Find the co-ordinates of A, B and C. Also, find the length of AB.
(ii) Find the equation of the circle with centre C which passes through A and B. Show that this circle also passes through the origin.
(iii) The tangent to the circle at B meets the straight line through 0 and C at the point D. Find the co-ordinates of D.

Can you also tell me how to work this out please?

Thank you, very much appreciated.

2. ## Response

Originally Posted by jevans
The line 3x + 4y = 24 cuts the x axis at point A and the y axis at point B. The point C is the mid-point of AB.

(i) Find the co-ordinates of A, B and C. Also, find the length of AB.
(ii) Find the equation of the circle with centre C which passes through A and B. Show that this circle also passes through the origin.
(iii) The tangent to the circle at B meets the straight line through 0 and C at the point D. Find the co-ordinates of D.

Can you also tell me how to work this out please?

Thank you, very much appreciated.
Okay, to find A, or where the line cuts the x-axis (x-intercept), set y = 0 in the line equation:

3x + 4*0 = 24
3x = 24
x = 8
You have the x-intercept coordinate: A = (8,0)

To find B, or where the line cuts the y-axis (y-intercept), set x = 0 in the line equation:

3*0 + 4y = 24
4y = 24
y = 6
You have the y-intercept coordinate: B = (0,6)

To find the midpoint C, use the formula:
((x1 + x2) / 2 , (y1 + y2) / 2)

From the two points above, it would be:
((8+0)/2 , (0+6)/2)
C = (4,3)

To find the distance, use this formula:

d = sqrt((0-8)^2 + (6-0)^2)
d = sqrt(64 + 36)
d = sqrt(100)
d = 10 units

3. Originally Posted by jevans
The line 3x + 4y = 24 cuts the x axis at point A and the y axis at point B. The point C is the mid-point of AB.

(i) Find the co-ordinates of A, B and C. Also, find the length of AB.
(ii) Find the equation of the circle with centre C which passes through A and B. Show that this circle also passes through the origin.
(iii) The tangent to the circle at B meets the straight line through 0 and C at the point D. Find the co-ordinates of D.

Can you also tell me how to work this out please?

Thank you, very much appreciated.
(ii) The equation used is the standard equation that has the form

(x - h)2 + (y - k)2 = r2
where h and k are the x- and y-coordinates of the center of the circle and r is the radius.

So, in using the point C = (4,3), and since the distance, which is also the diameter of the circle, we can find this equation. Note, we need the radius, so divide the distance by 2, to get 5:

We now have the form for the circle:
(x-4)^2 + (y-3)^2 = 25

I'm guessing to show that the circle passes through the origin, you would substitute x = 0 and y = 0 into the circle equation and make sure it checks out:

(0-4)^2 + (0-3)^2 = 25
16 + 9 = 25
25 = 25

4. ## Question for (iii)

Originally Posted by jevans
The line 3x + 4y = 24 cuts the x axis at point A and the y axis at point B. The point C is the mid-point of AB.

(i) Find the co-ordinates of A, B and C. Also, find the length of AB.
(ii) Find the equation of the circle with centre C which passes through A and B. Show that this circle also passes through the origin.
(iii) The tangent to the circle at B meets the straight line through 0 and C at the point D. Find the co-ordinates of D.

Can you also tell me how to work this out please?

Thank you, very much appreciated.
When you say meets the straight line through 0 and C at the point D. is the 0 meaning x = 0 or y = 0

5. Originally Posted by ajj86
When you say meets the straight line through 0 and C at the point D. is the 0 meaning x = 0 or y = 0
To be honest I'm not sure. It could mean the origin? If you don't know what it means, that's fine. Thanks for all your help.

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# if the line 24=3x 4y cuts the x axis

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