**Problem 1:**

If one of the straight lines given by the equation coincides with one of those given by , and the other lines represented by them be perpendicular, prove that

**Solution 1:**

Let the lines represented by be and , so that

and …call this equation (A)

The lines represented by are and so that

and …call this equation (B)

From (A) and (B), we get

which in turn ,

which in turn

and from (A), again, we get

and this in turn , that is, it .

Similarly, from (B), we get . Hence, the required result follows.

**Problem 2:**

If the equation ** **represents two straight lines, prove that the equation of the third pair of straight lines passing through the points where these meet the axes is .

**Solution 2:**

Equation of the lines passing through the intersection of the given lines and axes is

, or

Since it represents a pair of straight lines

,

(since ).

Hence, the required equation of the lines is

**Problem 3:**

Show that the equation represents three straight lines through the origin such that one of them bisects the angle between the other two. Also find the equation of the lines perpendicular to the given lines through the origin.

**Solution 3:**

The given equation can be written as , or

, or

, or

,or

, , and

which gives three straight lines passing through the origin with slopes 45 degrees, 60 degrees, and 30 degrees respectively showing that bisects the angles between the other two.

Now, equations of the lines through the origin perpendicular to these lines are

, , so their joint equation is given by

or,

*That’s all, folks !*

*Nalin Pithwa.*

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