Find the ratios in which the spheres x^2+y^2+z^2=24 divides the line joining the points (-1,1,2) and (-4,4,8)
To find the ratios in which the spheres x^2 + y^2 + z^2 = 24 divides the line joining the points (-1,1,2) and (-4,4,8), we can use the concept of internal and external division.
Step 1: Find the direction ratios of the line
The direction ratios of the line joining the points (-1,1,2) and (-4,4,8) can be calculated by subtracting the coordinates of one point from the other:
Direction ratios = (-4 - (-1), 4 - 1, 8 - 2)
= (-3, 3, 6)
Step 2: Find the equation of the line
The equation of the line can be written using the parametric form as:
x = -1 - 3t
y = 1 + 3t
z = 2 + 6t
Step 3: Substitute the line equation into the sphere equation
Substituting the values of x, y, and z from the line equation into the sphere equation x^2 + y^2 + z^2 = 24 gives:
((-1 - 3t)^2) + ((1 + 3t)^2) + ((2 + 6t)^2) = 24
Step 4: Simplify the equation
Expanding and simplifying the equation gives:
10t^2 + 12t + 3 = 0
Step 5: Solve the quadratic equation
Solving the quadratic equation using the quadratic formula:
t = [-b ± √(b^2 - 4ac)] / (2a)
where a = 10, b = 12, and c = 3, we have:
t = [-12 ± √(12^2 - 4(10)(3))] / (2(10))
= [-6 ± √(144 - 120)] / 20
= [-6 ± √24] / 20
= [-6 ± 2√6] / 20
= (-3 ± √6) / 10
Step 6: Calculate the ratios
The ratios in which the spheres divide the line can be obtained by substituting the values of t into the line equation.
When t = (-3 + √6) / 10:
x = -1 - 3((-3 + √6) / 10) = (-4 + 3√6) / 10
y = 1 + 3((-3 + √6) / 10) = (1 - 3√6) / 10
z = 2 + 6((-3 + √6) / 10) = (12 + 6√6) / 10
When t = (-3 - √6) / 10:
x = -1 - 3((-3 - √6) / 10) = (-4 - 3√6) / 10
y = 1 + 3((-3 - √6) / 10) = (1 + 3√6) / 10
z = 2 + 6((-3 - √6) / 10) = (12 - 6√6) / 10
Therefore, the ratios in which the spheres x^2 + y^2 + z^2 = 24 divide the line joining the points (-1,1,2) and (-4,4,8) are:
1. At the point (-4 + 3√6, 1 - 3√6, 12 + 6√6), the ratio is [(x - (-4 + 3√6)) / (-1 - (-4 + 3√6))] = [(y - (1 - 3√6)) / (1 - (-4 + 3√6))] = [(z - (12 + 6√6)) / (2 - (12 + 6√6))].
2. At the point (-4 - 3√6, 1 + 3√6, 12 - 6√6), the ratio is [(x - (-4 - 3√6)) / (-1 - (-4 - 3√6))] = [(y - (1 + 3√6)) / (1 - (-4 - 3√6))] = [(z - (12 - 6√6)) / (2 - (12 - 6√6))].
To find the ratios in which the spheres divide the line joining the points, we need to find the points where the line intersects the sphere.
Step 1: Given Data
We are given the equation of the sphere: x^2 + y^2 + z^2 = 24
And we have two points on the line: A(-1, 1, 2) and B(-4, 4, 8)
Step 2: Find the Equation of the Line
To find the equation of the line, we can use the equation of a line in 3D given two points:
(x - x1) / (x2 - x1) = (y - y1) / (y2 - y1) = (z - z1) / (z2 - z1)
Substituting the values:
(x + 1) / (-4 - (-1)) = (y - 1) / (4 - 1) = (z - 2) / (8 - 2)
Simplifying further:
(x + 1) / (-3) = (y - 1) / 3 = (z - 2) / 6
Step 3: Substitute the Line Equation into the Sphere Equation
Substituting the equation of the line into the equation of the sphere:
((-3t - 1)^2/9) + ((3t + 1)^2/9) + ((6t + 2)^2/9) = 24
(-3t - 1)^2 + (3t + 1)^2 + (6t + 2)^2 = 216
Simplify the equation:
9t^2 + 6t + 1 + 9t^2 + 6t + 1 + 36t^2 + 24t + 4 = 216
54t^2 + 36t + 6 = 216
54t^2 + 36t - 210 = 0
Step 4: Solve the Quadratic Equation
To solve the quadratic equation, we can use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
Substituting the values:
t = (-(36) ± √((36)^2 - 4(54)(-210))) / (2(54))
Simplifying further:
t = (-36 ± √(1296 + 45360)) / 108
t = (-36 ± √(46656)) / 108
t = (-36 ± 216) / 108
Therefore, we have two solutions:
t1 = (-36 + 216) / 108 = 180 / 108 = 5/3
t2 = (-36 - 216) / 108 = -252 / 108 = -7/3
Step 5: Find the Ratios
To find the ratios, we substitute the values of t into the equation of the line:
For t1 = 5/3:
x = -1 + (-3)(5/3) = -1 - 5 = -6
y = 1 + 3(5/3) = 1 + 5 = 6
z = 2 + 6(5/3) = 2 + 10 = 12
Therefore, the point where the line intersects the sphere for t = 5/3 is P1(-6, 6, 12).
For t2 = -7/3:
x = -1 + (-3)(-7/3) = -1 + 7 = 6
y = 1 + 3(-7/3) = 1 - 7 = -6
z = 2 + 6(-7/3) = 2 - 14 = -12
Therefore, the point where the line intersects the sphere for t = -7/3 is P2(6, -6, -12).
Therefore, the ratios in which the spheres divide the line joining the points (-1,1,2) and (-4,4,8) are:
AP1 : P1B = |AP1| / |P1B|
AP2 : P2B = |AP2| / |P2B|
To find the lengths, we use the distance formula:
|AP1| = √((-6 - (-1))^2 + (6 - 1)^2 + (12 - 2)^2)
|P1B| = √((-4 - (-6))^2 + (4 - 6)^2 + (8 - 12)^2)
|AP2| = √((6 - (-1))^2 + (-6 - 1)^2 + (-12 - 2)^2)
|P2B| = √((-4 - 6)^2 + (4 - (-6))^2 + (8 - (-12))^2)
After calculating the distances, we can find the ratios by dividing the distances.
Note: The final ratios will depend on the exact values of the distances and need to be calculated.
Spheres?
Only one sphere divides that line.
The line goes through the origin (0,0,0). That is apprent by inspection. The sphere is centered at the origin. That is apparent from the equation of the sphere.
Therefore the line (if extended) must intersect the sphere at two points equidistant from the origin, at radius sqrt(24)=4.899. The first point is inside the sphere; the second is outside; therefore the line is divided into two parts only.
Find the point of intersection to complete the problem.