Question

The set of lines ax + by + c = 0 where 3a + 2b + 4c = 0 is concurrent at the point ___.

Answer 1

Hint: Convert 3a + 2b + 4c = 0 into the form ax + by + c = 0. The coefficients of x and y upon solving gives us the point of concurrence. Compare coefficients of x and y to get the desired point.

Complete step-by-step answer:
Given:
The set of lines ax + by + c = 0, where 3a + 2b + 4c = 0.

We have to rewrite the given equation in the form of ax + by + c = 0.
So divide the equation 3a + 2b + 4c = 0 by 4 to make the coefficient of c equal to 1.

That gives us $\dfrac{3}{4}$a + $\dfrac{2}{4}$b + c = 0

Now, by comparing the coefficients of x and y, (which is ‘a’ and ‘b’ in our case). We get the point of concurrency.
Thus, the point of concurrency is ($\dfrac{3}{4}$,$\dfrac{1}{2}$).

Note: In such problems the key is to compare the coefficients.
A point of concurrency is a place where three or more, but at least three lines, rays, segments or planes intersect in one point. If they do, then those lines are considered concurrent, or the rays are considered concurrent and the point at which they intersect is called the point of concurrence.
Here the set of lines is ax + by + c = 0 and the given property is 3a + 2b + 4c = 0 which helps solve the problem.