Database - Equation of a Circle
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Question
Find the equation of the circle which passes through A(8, 1) and B(7, 0) and has, for its tangent at B, the line 3x - 4y - 21 = 0
Answer
Step 1 : Rearrange the line 3x - 4y - 21 = 0 into the form of y = mx + c
4y = 3x - 21
y = (3/4)x -21/4
Hence, m = 3/4
Step 2 Find the gradient of BC where B(7,0) and centre C(-g, -f)
m1 = [0-(-f)]/[7-(-g)]
= f/[7+g]
Step 3 Since the line 3x - 4y - 21 = 0 is tangent to the circle at point B, so the line
3x - 4y = 0 is perpendicular to BC ie m = - 1/m1
3/4 = - (7+g)/f
f = - (4/3)(7 + g) ------------------ (0)
Step 4 Since the points A(8,1) and B(7,0) are points on the circle, substitute them
into the equation of the circle to form 2 simultaneous equations
Equation of the circle is x^2 + y^2 +2gx +2fy + c = 0
Substitute point A (8,1), 8^2 + 1^2 + 2g(8) + 2f(1) + c = 0
65 + 16g + 2f + c = 0 ------------- (1)
Substitute point B (7, 0), 7^2 + 0^2 + 2g(7) + 2f(0) + c = 0
49 + 14g + c = 0 ------------------ (2)
Equation (2) - Equation (1)
16 + 2g + 2f = 0
8 + g + f = 0
f = - 8 - g ---------------------------- (3)
Step 5 Susbtitute equation (3) into equation (0) to solve for g
Substitute f = - 8 - g into equation (0)
- 8 - g = - (4/3)(7 + g)
- 24 - 3g = - 28 - 4g
g = - 4
Step 6 Substitute g = - 4 into equation (3) to solve for f
f = - 8 - (- 4)
f = - 4
Step 7 Substitute g = - 4 into equation (2) to solve for c
49 + 14 (- 4) + c = 0
c = 7
Step 8 Substitue g = - 4 , f = - 4 and c = 7 into the equation of the circle
x^2 + y^2 +2gx +2fy + c = 0
x^2 + y^2 + 2(- 4)x + 2(- 4)y + 7 = 0
x^2 + y^2 - 8x - 8y + 7 = 0
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wow this qn is tough. I really dont think they wld test too hard since this is new in the syllabus.
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Originally posted by davidche:
wow this qn is tough. I really dont think they wld test too hard since this is new in the syllabus.
Dear Davidche,
The two questions that I posted in the thread " Must know new 2008 Add
Questions" are actual past "O" level add maths exam questions with slight
modification.
The questions on the equation of a circle (in the mid 60s and early 70s) in the past
"O" level add maths do test on tangent, passes through two points, circle touches
the axis and so on.
Perhaps, this is the first year, the "O" level examiners will test on the easier
question on the equation of a circle first.
Regards,
ahm97sic
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Originally posted by Ahm97sic:
Dear Davidche,
The two questions that I posted in the thread " Must know new 2008 Add
Questions" are actual past "O" level add maths exam questions with slight
modification.
The questions on the equation of a circle (in the mid 60s and early 70s) in the past
"O" level add maths do test on tangent, passes through two points, circle touches
the axis and so on.
Perhaps, this is the first year, the "O" level examiners will test on the easier
question on the equation of a circle first.
Regards,
ahm97sic
I am relieved that you said this XD
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Dear Davidche,
Keep calm and don't panic is most important when you are taking the exam.
Do those questions that you can do easily first and avoids carelessness, once you realise the numbers of questions that you can solve, you will be more confident and do well in th exam.
Even if the exam is difficult, don't panic. Do parts of the question that you can do and get some marks here and there, you can still pass the exam. So, keep calm and don't panic in any exam.
Have enough sleep, otherwise it will affect your performance during the exam.
All the best for your add maths exam tomorrow.
Regards,
ahm97sic
PS : Please come to my forums more often. OK.
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Haha i will. And yeah i sleep at mid night kinda late but I wake up at 9.30. Shld be ok lar. 9 hours.
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Dear Davidche,
Indeed, enough sleep is important for the best performance in exam.
All the best for your add maths exam.
Warmest Regards,
ahm97sic