Consider the plane P with equation −2x−2y−z = −11 and line L with equations
x = −3−4t
y = −2−3t
z = −6−4t
Find two distinct points Q1 and Q2 on the line L at distance 3 to the plane P.
Q1 = (0, 0, 0)
Q2 = (0, 0, 0)
 👍
 👎
 👁
 ℹ️
 🚩
1 answer

Let Q be a general point on the line
Q is (34t, 23t, 64t)
Plane : 2x 2y z + 11 = 0
Using the distance from a point to a plane formula
 2(34t)  2(23t)  (64t) + 11/ √(4+4+1) = 3
± (6 + 8t + 4 + 6t + 6 + 4t + 11)/3 = 3
using the positive
18t + 27 = 9
18t = 18
t = 1
then Q is (3+4, 2+3, 6+4) ,
Q1 is (1, 1, 2)
using the negative result from above:
(6 + 8t + 4 + 6t + 6 + 4t + 11)/3 = 3
18t  27 = 9
18t = 36
t = 2
Q2 is (3+8, 2+6, 6+8)
= Q2(5, 4, 2)
check:
midpoint of Q1Q2 = (3, 5/2, 0) , this point should lie on our plane
LS = 2(3) 2(5/2)  (0)
= 6 5  0 = 11
RS = 11
looks good! 👍
 👎
 ℹ️
 🚩
mathhelper
Answer this Question
Related Questions

Algebra
1. for the data in the table, does y vary directly with x? if it does write an equation for the direct variation. 10  12 15  18 20  24 A. yes; y = 1.2x B. yes; y = 2x C. yes; y = x + 1 D. no; y does not vary directly with x 2. For the data in the table,

Math (Calculus) (mean value theorem emergency)
Consider the graph of the function f(x)=x^2x12 a) Find the equation of the secant line joining the points (2,6) and (4,0). I got the equation of the secant line to be y=x4 b) Use the Mean Value Theorem to determine a point c in the interval (2,4)

Math
1. Determine the scalar equation of the plane with vector equation Vector r= (3,1,4) +s(2,1,5) + t(3,2,2). 2. Determine the value of k so that the line with parametric equations x = 2 + 3t, y = 2 + 5t, z = kt is parallel to the plane with equation 4x

can you answer this very quick
On a graph, the solution to a system of linear equations will be represented by ______. A. the xintercept of each line B. the yintercept of each line C. all the points on either line D. the intersection of the lines

algebra
1.What is the slope of the function shown on the graph? A line is graphed on a four quadrant coordinate plane. The xaxis and the yaxis go from negative 10 to 10 in increments of 1. The line passes through (0, 7) and (4, 2). A−56 B−54 C−65 D−45

Chemistry
Write balanced chemical equations for all transferals of the compounds from organic to aqueous phases and for all precipitation reactions. **Note: If a compound does not move from one layer to another, no reaction has occurred, and no equation can be

Math
Lesson 8: Systems of Linear and Quadratic Equations Check my work 1. Solve the system of equations. y = 2x^2  3 y = 3x  1 a. no solution b. (1/2, 5), (2, 5/2) c. (1/2, 5/2), (2,5)*** d. (1/2, 5/2), (2, 5) 2.how many real number solutions does the

math
A particle moves in the xyplane so that its position for t ≥ 0 is given by the parametric equations x = ln (t + 1) and y = k(t^2) , where k is a positive constant. The line tangent to the particle’s path at the point where t = 3 has slope 8. What is

algebra
Write an equation of the line containing the given point and perpendicular to the given line. express your answer in the fory y=mx+b. (8,9); 7x+y=2 the equation of the line is y=?

maths
P(6,3), Q(3,7), and R(4,2) are three points in a plane. A is the midpoint of QR and B is the foot of the perpendicular from Q to PR. Find; (A) the cordinates of A; (B) the equations of the lines PA and QB; (C) the point of intersection of the lines PA and

Velocity and Displacement
A plane is sitting on a runway, awaiting takeoff. On an adjacent parallel runway, another plane lands and passes the stationary plane at a speed of 47 m/s. The arriving plane has a length of 32 m. By looking out the window (very narrow), a passenger on the

Algebra 1
1. What is the domain of the function {(18,34),(22,64),(94,36),(11,18),(91,45)}? 8.what is the equation of the line that includes the point (4,3) and has a slope of 2? 12.what is the solution to the following system of equations? Y=2x4 y=x 15.which

Geometry
1.) A skateboard ramp has a 158° entry angle. Find the measure of ∠1. 2.) Which of the following equations is parallel to the line passing through A(2, 7);B(2, 2)? 3.) Which of the following equations would graph a line parallel to 3y = 2x  x + 5?

Calculus
a) Find a Cartesian equation relating and corresponding to the parametric equations: x=2sin(3t), y=9cos(3t). Write your answer in the form P(x,y)=0, where P(x,y) is a polynomial in x and y, such that the coefficient of y^2 is 4. b)Find the equation of the

Algebra
Please Check My Anwsers What is the slope of the line in the graph show below? /content/media/390571112201142500PM1076804072.png A) 2 B) 1 C) 1 (My Choice) D) 2 Find the slope of a line that passes through (–2, –3) and (1, 1) A) 1/3 B) 1 C) 2 (My

Calculus
Find symmetric equations for the line. (Use the parameter t.) The line through (−8, 4, 7) and parallel to the line x/2= y/3= z+1

Calculus
Find the equation of the plane through (1,2,2) that contains the line x = 2t, y = 3  t, z = 1 +3t

Geometry
Okay heres the pic. There is a plane with points B,D, and E.those [oints are collinear in the plane. And also outside of the plane is points A, and C. and those points are collinear outside of the plane.thought Line AC intersect at point B of the line

Calc.
Determine the value of k so that the line with parametric equations x = 2 + 3t, y = 2 + 5t, z = k is parallel to the plane with equation 4x + 3y – 3z 12 = 0

Math
Write an equation for the line that is parallel to the given line and passes through the given point. y=3x+4;(6,8) A. y=3x10 B. y=3x+6 C. y=3x+6 D. y=1/3x6 Tell whether the lines for each pair of equations are parallel, perpendicular, or neither.
Still need help?
You can ask a new question or browse existing questions.