Let the line L pass through (1, 1, 1) and intersect the lines fracx - 12 = fracy + 13 = fracz - 14$\frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z - 1}{4}$ and fracx - 31 = fracy - 42 = fracz1$\frac{x - 3}{1} = \frac{y - 4}{2} = \frac{z}{1}$ . Then, which of the following points lies on the line L?

### Related Formula General coordinates of any variable point on a 3D line given symmetric form equations: P_1 = (2lambda + 1, \, 3lambda - 1, \, 4lambda + 1)$$P_1 = (2\lambda + 1, \, 3\lambda - 1, \, 4\lambda + 1)$$ P_2 = (mu + 3, \, 2mu + 4, \, mu)$$P_2 = (\mu + 3, \, 2\mu + 4, \, \mu)$$ ### Core Logic Let line L intersect Line 1 at point A(2lambda + 1, 3lambda - 1, 4lambda + 1)$A(2\lambda + 1, 3\lambda - 1, 4\lambda + 1)$ and Line 2 at point B(mu + 3, 2mu + 4, mu)$B(\mu + 3, 2\mu + 4, \mu)$. Since line L passes through C(1, 1, 1)$C(1, 1, 1)$, the direction ratios computed from vector segment AC$AC$ must be proportional to the direction ratios computed from vector segment BC$BC$. ### Step 1: Determine Direction Ratio Parameters

Direction ratios of AC$AC$ segment: vecAC = (2lambda + 1 - 1, \, 3lambda - 1 - 1, \, 4lambda + 1 - 1) = (2lambda, \, 3lambda - 2, \, 4lambda)$$\vec{AC} = (2\lambda + 1 - 1, \, 3\lambda - 1 - 1, \, 4\lambda + 1 - 1) = (2\lambda, \, 3\lambda - 2, \, 4\lambda)$$ Direction ratios of BC$BC$ segment: vecBC = (mu + 3 - 1, \, 2mu + 4 - 1, \, mu - 1) = (mu + 2, \, 2mu + 3, \, mu - 1)$$\vec{BC} = (\mu + 3 - 1, \, 2\mu + 4 - 1, \, \mu - 1) = (\mu + 2, \, 2\mu + 3, \, \mu - 1)$$ Equating directional proportionality ratios: fracmu + 22lambda = frac2mu + 33lambda - 2 = fracmu - 14lambda$$\frac{\mu + 2}{2\lambda} = \frac{2\mu + 3}{3\lambda - 2} = \frac{\mu - 1}{4\lambda}$$ ### Step 2: Solve for Parameter Intersection values From the first and third fractional groups: fracmu + 22lambda = fracmu - 14lambda implies 2(mu + 2) = mu - 1$$\frac{\mu + 2}{2\lambda} = \frac{\mu - 1}{4\lambda} \implies 2(\mu + 2) = \mu - 1$$ 2mu + 4 = mu - 1 implies mu = -5$$2\mu + 4 = \mu - 1 \implies \mu = -5$$ Substitute mu = -5$\mu = -5$ back into the second parameter group linkage to evaluate the target structural direction indicators, which gives the simplified direction ratio vector for BC$BC$ as: textD.R.s = (-5 + 2, \, 2(-5) + 3, \, -5 - 1) = (-3, \, -7, \, -6) equiv (3, \, 7, \, 6)$$\text{D.R.s} = (-5 + 2, \, 2(-5) + 3, \, -5 - 1) = (-3, \, -7, \, -6) \equiv (3, \, 7, \, 6)$$ ### Step 3: Construct Line Equation and Verify Choice Equation of line L passing through C(1, 1, 1)$C(1, 1, 1)$ with direction vector (3, 7, 6)$(3, 7, 6)$: fracx - 13 = fracy - 17 = fracz - 16$$\frac{x - 1}{3} = \frac{y - 1}{7} = \frac{z - 1}{6}$$ Let's check option (7, 15, 13)$(7, 15, 13)$: frac7 - 13 = frac63 = 2$$\frac{7 - 1}{3} = \frac{6}{3} = 2$$ frac15 - 17 = frac147 = 2$$\frac{15 - 1}{7} = \frac{14}{7} = 2$$ frac13 - 16 = frac126 = 2$$\frac{13 - 1}{6} = \frac{12}{6} = 2$$ Since all values match perfectly, (7, 15, 13)$(7, 15, 13)$ lies on the line L. ### Pattern Recognition By comparing the first and third fractional terms containing lambda$\lambda$ in the denominator, you can solve for mu$\mu$ independently without tracking complex cross-multiplied quadratic lambdamu$\lambda\mu$ variations. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Mathematics: Three Dimensional Geometry