The straight line 3x+y+4=0 intersects the circle x^2 + y^2 -2x-6y-10=0 at two points A and B. Find the length of the chord AB without finding the coordinate of A and B.
I have thought a method, that is finding the half length of AB. I need to find
the length between the centre and the mid point of AB first but I don't know how to find it. Please teach me how to find it or is then any methods other than the
method above? Thank you!!
F.5 Maths Equation of circle
2015-03-06 6:54 am
回答 (2)
2015-03-06 12:37 pm
✔ 最佳答案
DenoteO as the centre of the circle, and M be the mid-point of AB.Centre of the circle = (-(2)/2, -(-6)/2) = (1, 3)
Radius of the circle, OA = √[1² + 3² -(-10)] = √20
OM = |3(1) + (3) + 4| / √(3² + 1²)(Pythagorean theorem)
OM = √10
In ΔOMA :
∠OMA = 90°
OM² + AM² =OA² (Pythagorean theorem)
(√10)² + AM² = (√20)²
AM² = 10
AM = √10
AB = 2 AM
AB = 2√10
2015-03-06 8:10 am
At first, you have to find the centre of circle, O = (-D/2,-E/2) = (1,3)
Radius of the circle, R = √ (-2/2)^2 +(-6/2)^2 -(-10) =√20
From the equation of the line AB ：3x+y+4=0 , the slope m=-3
Draw the line from centre prependicular to intersect AB at N,
the slope of ON = 1/3 (Because slope ON x slope of m = -1)
You can find the equation of ON, y-3 = 1/3(x-1)
Solve, 3x+y+4=0 -------(1)
y-3 = 1/3(x-1) --------(2)
After solving (1) and (2), you will get
x= -3 , y=2
∴ N= (-3,2)
ON =√ (-3-1)^2 + (2-3)^2 =√17
OA = Radius = √ 20
(ON)^2 + (AN)^2 = (OA)^2
(√17)^2 + (AN)^2= (√ 20)^2
(AN)^2= (√ 20)^2 - (√17)^2 (Can you find AN=?)
Since AB = 2AN
then AB = ？ (calculate it by yourself please!)
2015-03-06 00:17:00 補充：
你試試再完成餘下部份，有問題再找我。
Radius of the circle, R = √ (-2/2)^2 +(-6/2)^2 -(-10) =√20
From the equation of the line AB ：3x+y+4=0 , the slope m=-3
Draw the line from centre prependicular to intersect AB at N,
the slope of ON = 1/3 (Because slope ON x slope of m = -1)
You can find the equation of ON, y-3 = 1/3(x-1)
Solve, 3x+y+4=0 -------(1)
y-3 = 1/3(x-1) --------(2)
After solving (1) and (2), you will get
x= -3 , y=2
∴ N= (-3,2)
ON =√ (-3-1)^2 + (2-3)^2 =√17
OA = Radius = √ 20
(ON)^2 + (AN)^2 = (OA)^2
(√17)^2 + (AN)^2= (√ 20)^2
(AN)^2= (√ 20)^2 - (√17)^2 (Can you find AN=?)
Since AB = 2AN
then AB = ？ (calculate it by yourself please!)
2015-03-06 00:17:00 補充：
你試試再完成餘下部份，有問題再找我。
收錄日期: 2021-04-15 18:25:48
原文連結 [永久失效]:
https://hk.answers.yahoo.com/question/index?qid=20150305000051KK00137