# Answer in Geometry for Vanel #216981

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

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.