Question

In the cartesian plane OXY, we consider the lines with equations

ax+3y+4=0 and x+2ay+7=0 with a as real parameter. Which of the following statements is correct?

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

B. A unique value of a exist 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-aligne points in the plane. How many lines can one find that are exactly at the same distance from these three points?

Answer 1

When two lines are parallel, then their gradients m1 and m2 must 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 appropriate statement 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

Answer:

The three non-collinear points can be from a triangle MNO, where M, N, and O form a triangle. There exist only one point at the center that is. lines joining the mid point to the opposite vertex, which 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.