Question

In the cartesian plane OXY, we consider the lines with equation ax + 3y + 4=0 and x + 2ay + 7=0 with a as real parameter. Which of the following statements is true?

A. There exist a unique value for a for which the lines are parallel and distinct

B. A unique value of a exists for which the lines are coincident

C. Two values of a exist for which the lines are parallel

D. No value of a for which the lines are parallel

We consider 3 non-aligned points in the plane.  How many lines can one find that are

 exactly at the  same distance from these three points

Answer 1

If the two lines are parallel, then their slopes m1 and m2 should be equal.

Therefore,

ax + 3y + 4 = 0

$y = \frac{-ax}{3} - \frac{-4}{3}$

and x + 2ay + 7 =0

$y = \frac{-1}{2a} - \frac{-7}{2}$

$m1 = \frac{-a}{3}$

$m2 = \frac{-1}{2a}$

Putting m1 = m2, we get

$$a = +\sqrt{\smash[b]{3/2}}$$

and, $a = -\sqrt{\smash[b]{3/2}}$

Therefore, the the correct option is:

C. Two values of a exist for which the lines are parallel

We consider 3 non-aligned points in the plane.  How many lines can one find that are

 exactly at the  same distance from these three points

The three non-colinear points can be from a triangle ABC, where A, B, and C form a triangle. There can be only one point at the center i.e. lines joining the mid point to the opposite vertex, that is equidistant from all three. Therefore, We consider 3 non-aligned points in the plane, then there can be 3 lines that can be equidistance from 3 non-aligned points in the plane.

Final Answer: There can be 3 lines that can be equidistance from 3 non-aligned points in the plane.