Straight Lines
Distance Measured Parallel to a Line
nta_pyq_2024_jan
Grade 11
Question:
The distance of the point $(2,3)$ from the line $2x-3y+28=0$, measured parallel to the line $\sqrt{3}x-y+1=0$, is equal to
$4\sqrt{2}$
$6\sqrt{3}$
$3+4\sqrt{2}$
$4+6\sqrt{3}$
Step-by-Step Solution
Key Concept: Line parallel to $\sqrt3x-y+1=0$ through $(2,3)$: parametric form with $\tan\theta=\sqrt3\Rightarrow\cos\theta=1/2,\sin\theta=\sqrt3/2$. Point $P=(2+r/2,3+r\sqrt3/2)$. Substitute into $2x-3y+28=0$.
$r=4+6\sqrt3$.
Correct Answer: 4