Let the values of $\lambda$ for which the shortest distance between the lines $\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}$ and $\frac{x - \lambda}{3} = \frac{y - 4}{4} = \frac{z - 5}{5}$ is $\frac{1}{\sqrt{6}}$ be $\lambda_1$ and $\lambda_2$. Then the radius of the circle passing through the points (0, 0), $(\lambda_1, \lambda_2)$ and $(\lambda_2, \lambda_1)$ is

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 $A_1(1, 2, 3)$ and has direction vector $\vec{b}_1 = 2\hat{i} + 3\hat{j} + 4\hat{k}$. The second line passes through point $A_2(\lambda, 4, 5)$ and has direction vector $\vec{b}_2 = 3\hat{i} + 4\hat{j} + 5\hat{k}$.

Step 2: Calculate shortest distance components

Next, we calculate the vector connecting the two points, $\vec{a}_2 - \vec{a}_1$, and the cross product of the direction vectors, $\vec{b}_1 \times \vec{b}_2$. We also find the magnitude of the cross product, which is needed for the shortest distance formula.

Step 3: Solve for $\lambda$

We use the formula for the shortest distance between two skew lines. Substituting the calculated values and the given shortest distance $\frac{1}{\sqrt{6}}$, we solve for $\lambda$. This gives us two possible values for $\lambda$.

Step 4: Determine $\lambda_1$ and $\lambda_2$

From the equation $|-\lambda + 3| = 1$, we have two cases: $-\lambda + 3 = 1$ or $-\lambda + 3 = -1$. Solving these equations yields $\lambda_1 = 2$ and $\lambda_2 = 4$.

Step 5: Identify points for the circle

The three points through which the circle passes are $(0, 0)$, $(\lambda_1, \lambda_2) = (2, 4)$, and $(\lambda_2, \lambda_1) = (4, 2)$.

Step 6: Find the equation of the circle

The general equation of a circle is $x^2 + y^2 + 2gx + 2fy + c = 0$. 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 $R = \sqrt{g^2 + f^2 - c}$. Substituting the values, we get the radius.

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• Find the equation of the plane passing through the point (1, 2, -3) and perpendicular to the planes x + 2y + 3z = 4 and 2x - 3y + 4z = 5.
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