Question

The symmetric form of the equation of the line \(x = ay+b\), \(z = cy+d\) is

• A. \(\frac{x-a}{b} = y = \frac{z-c}{d}\)
• B. \(\frac{x-b}{a} = \frac{y-d}{c} = z\)
• C. \(x = \frac{y-a}{b} = \frac{z-c}{d}\)
• D. \(\frac{x-b}{a} = y = \frac{z-d}{c}\)

Hint:

Solve each equation for y and equate.

Answer 1

The Correct Option is D

Step 1: Parametrise:
Let $y=t$. Then $x=at+b$ and $z=ct+d$.

Step 2: Point and Direction:
At $t=0$ the point is $(b,0,d)$. The direction vector is $(a,1,c)$.

Step 3: Symmetric Form:
$\dfrac{x-b}a=\dfrac{y-0}1=\dfrac{z-d}c=t$. Option (D).

Final Answer:
Option (D). \[ \boxed{\text{(D) } \frac{x-b}{a}=y=\frac{z-d}{c}} \]