Question 11 of 25intermediate🔧 ApplyShort Answer4 marks

Find the cartesian equation of the line which passes through the point (−2,4,−5)(-2, 4, -5) and parallel to the line given by x+33=y−45=z+86\frac{x+3}{3} = \frac{y-4}{5} = \frac{z+8}{6}.

Correct Answer

The cartesian equation of the required line is x+23=y−45=z+56\frac{x + 2}{3} = \frac{y - 4}{5} = \frac{z + 5}{6}

Exercise: EXERCISE 11.2 | Q: 6 | (Page Page 13)
For More Understanding

Explanation

This question tests understanding of parallel lines in 3D space. When two lines are parallel, they have the same direction ratios. The given line has direction ratios (3, 5, 6) extracted from the denominators. The new line must use these same direction ratios but pass through the given point (-2, 4, -5). Using the standard Cartesian form formula from the textbook, we substitute the point coordinates and direction ratios to get the final equation.

Solution Steps

  1. Step 1: From the given line equation x+33=y−45=z+86\frac{x+3}{3} = \frac{y-4}{5} = \frac{z+8}{6}, identify the direction ratios as (a,b,c)=(3,5,6)(a, b, c) = (3, 5, 6).

  2. Step 2: Since the required line is parallel to the given line, it will have the same direction ratios (3,5,6)(3, 5, 6).

  3. Step 3: The required line passes through the point (x1,y1,z1)=(−2,4,−5)(x_1, y_1, z_1) = (-2, 4, -5).

  4. Step 4: Using the Cartesian equation form x−x1a=y−y1b=z−z1c\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}, substitute the values.

  5. Step 5: x−(−2)3=y−45=z−(−5)6\frac{x - (-2)}{3} = \frac{y - 4}{5} = \frac{z - (-5)}{6} simplifies to x+23=y−45=z+56\frac{x + 2}{3} = \frac{y - 4}{5} = \frac{z + 5}{6}.

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