Write equation in symmetric form (x-x₁)/a = (y-y₁)/b = (z-z₁)/c.

Equation of Line Through Two Points in 3D

Question

Find the equation of the line passing through the points $A(1, 2, 3)$ and $B(4, 6, 9)$ in symmetric form.

Solution — Step by Step

Subtract coordinates of $A$ from $B$ :

\vec{b} - \vec{a} = (4-1,\ 6-2,\ 9-3) = (3,\ 4,\ 6)

This vector tells us the direction the line “travels” — every step along the line shifts $x$ by 3, $y$ by 4, $z$ by 6.

Use point $A(1, 2, 3)$ as the fixed point and $(3, 4, 6)$ as direction ratios:

\frac{x - 1}{3} = \frac{y - 2}{4} = \frac{z - 3}{6}

This is the answer in symmetric form. Any point $(x, y, z)$ on the line satisfies this equation.

Plug $B(4, 6, 9)$ :

\frac{4-1}{3} = \frac{6-2}{4} = \frac{9-3}{6} = 1

All three ratios equal 1, so $B$ is on the line. Always do this 30-second check in board exams — it costs nothing and saves marks.

Why This Works

The symmetric form is essentially saying: the ratio of displacement from the fixed point to the direction ratio is the same for all three coordinates. That common ratio is the parameter $\lambda$ — set each fraction equal to $\lambda$ and you get the parametric form.

When $\lambda = 0$ , you land on $A$ . When $\lambda = 1$ , you land on $B$ . For any real $\lambda$ , you land on some point on the line. This connection between symmetric and parametric form is exactly what CBSE asks you to convert between.

The direction ratios $(3, 4, 6)$ don’t have to be simplified or converted to direction cosines for the equation — direction ratios work perfectly as denominators.

Alternative Method

You can use point $B$ instead of $A$ as the fixed point:

\frac{x - 4}{3} = \frac{y - 6}{4} = \frac{z - 9}{6}

Both equations represent the same line. This trips up students who think there’s only one correct form. If the denominators (direction ratios) are proportional and the point lies on the line, the equation is valid.

In JEE Main, if two answer options show the line with different fixed points but the same direction ratios, both are correct. Check if the fixed point satisfies the other equation — if yes, they’re the same line.

Common Mistake

Reversing the subtraction. Students sometimes write direction ratios as $(1-4, 2-6, 3-9) = (-3, -4, -6)$ . This gives:

\frac{x-1}{-3} = \frac{y-2}{-4} = \frac{z-3}{-6}

This is still the same line — just traversed in the opposite direction. But in a board exam, if the expected answer uses $(3, 4, 6)$ , you’ll lose marks for sign errors. Always subtract $A$ from $B$ (head minus tail) to match the standard expected form.

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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