Let the values of λ for which the shortest distance between the lines 2x−1=3y−2=4z−3 and 3x−λ=4y−4=5z−5 is 61 be λ1 and λ2. Then the radius of the circle passing through the points (0, 0), (λ1,λ2) and (λ2,λ1) is
Get the complete, step-by-step math solution for: "Let the values of for which the shortest distance between the lines (x-1)/(2) = (y-2)/(3) = (z-3)/(4) and (x - )/(3) = (y - 4)/(4) = (z - 5)/(5) is {1...". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Identify line parameters
First, we identify the position vectors and direction vectors for the two given lines. The first line passes through point A1(1,2,3) and has direction vector b1=2i^+3j^+4k^. The second line passes through point A2(λ,4,5) and has direction vector b2=3i^+4j^+5k^.
Step 2: Calculate shortest distance components
Next, we calculate the vector connecting the two points, a2−a1, and the cross product of the direction vectors, b1×b2. We also find the magnitude of the cross product, which is needed for the shortest distance formula.
Step 3: Solve for λ
We use the formula for the shortest distance between two skew lines. Substituting the calculated values and the given shortest distance 61, we solve for λ. This gives us two possible values for λ.
Step 4: Determine λ1 and λ2
From the equation ∣−λ+3∣=1, we have two cases: −λ+3=1 or −λ+3=−1. Solving these equations yields λ1=2 and λ2=4.
Step 5: Identify points for the circle
The three points through which the circle passes are (0,0), (λ1,λ2)=(2,4), and (λ2,λ1)=(4,2).
Step 6: Find the equation of the circle
The general equation of a circle is x2+y2+2gx+2fy+c=0. Since the circle passes through (0,0), we find that c=0. Substituting the other two points, (2,4) and (4,2), we get a system of two linear equations in g and f.
Step 7: Solve for g and f, and calculate radius
Solving the system of equations g+2f=−5 and 2g+f=−5, we find g=−5/3 and f=−5/3. The radius of the circle is given by the formula R=g2+f2−c. Substituting the values, we get the radius.