Question

Write the condition for no solution and infinite solutions for two lines.

Answer 1

Hint: We will solve the problem mathematically as well as practically. We will take the general equation of two lines; understand the mathematical relation between the coefficients of the equation for no solution and infinite solution.

Complete step-by-step answer:
We know that the general equation of a line is $ax+by+c=0$
Where, x and y are the variables and we know that a, b and c are the constants.
Let us take two lines according to the general equation with different coefficients and see the relation between them when there is no solution and infinite solution.
Let the lines are:
$px+qy+r=0$ and $ux+vy+w=0$ .
In the given two lines x and y are the variables and the terms p, q, r, u, v and w.
First we have to understand that the solution between the lines is given if and only if the lines intersect and the point of intersection is the solution of the lines.
For no solution,
Two lines have no solution, if these two lines are parallel to each other.
The lines are parallel to each other means that the slopes of the lines are equal.
Mathematical representation of the equation with no solution for the given set of general lines is:
${\displaystyle \frac{p}{u}}={\displaystyle \frac{q}{v}}\ne {\displaystyle \frac{r}{w}}$
For infinite solution,
Two lines have infinite solutions, if these two lines are concurrent.
The lines are concurrent to each other means that the slopes of the lines are equal as well as equal to the ratio of the constant term.
Mathematical representation of the equation with infinite solution for the given set of general lines is:
${\displaystyle \frac{p}{u}}={\displaystyle \frac{q}{v}}={\displaystyle \frac{r}{w}}$
Hence, the given sets of conditions are valid for equations with no solution and infinite solution.


Note: In order to solve the problem related to coordinate geometry students must not solve the problem mathematically but should also try to find out the physical meaning of the problem. Students must remember these conditions in order to solve such problems of coordinate geometry.