Enough of the boring theory.. time for some examples !
Example 1 Find the equation of the line which
(i) has slope 3, and y-intercept 4
(ii) passes through a point on the Y-axis, 5 units below the origin, and makes an angle of 45o with the X-axis
Solution Easy ones. You’ll get the answers by just plugging in the values.
(i) The slope, m is given. The y-intercept, c is given. And, we know the slope-intercept form of the equation, y = mx + c. Nothing much to do here then. The equation will be y = 3x + 4.
(ii) Here, “passes through a point on the Y-axis, 5 units below the origin” is a fancy way to say that the y-intercept is -5
and.. “makes an angle of 45o with the X-axis” implies that the slope is tan45o which equals 1
By putting in the values, we get the equation as y=1x+(-5) or y=x-5
Example 2 A line having slope m, passes through the point (0, 4), and makes a triangle of area 8 sq units with the coordinate axes. Find the value of m, and the equation of the line.
It is not necessary, but generally helpful if you’re able to get an idea about how the given line (or a curve in general) should look. There might be certain cases in which there may be more than one curve which satisfies a given set of conditions, and solving only through equations might not make it obvious.
Here’s how the line could look…
Notice there are two possibilities. The line can form two triangles as shown (OAC and OBA) with same areas.
Let the equation of the line be y=mx+c. Using the given conditions, we know the value of c, which is 4. We need to find ‘m’.
Given that the area of the triangle formed by the line with the axes is 8 square units, our aim is to express this area in terms of ‘m’ and find its value.
Ar(OAC) = 1/2 x OC x OA. Now, tanθ (or m) = OA / OC, making OC = OA / m. Therefore Ar(COA) = 1/2 x OA/m x OA
We’re done. Substituting in the values (i.e. Ar(OAC) = 8, and OA = 4), we get m = 1.
Hence, the equation to the line is given by y = x + 4.
What about the other possibility when the line forms the triangle OAB?
In this case, the difference is that tan(π-θ) = OA / OB (in triangle OAB). After performing the same calculations as above, we’ll get m = -1, giving a different equation as y = -x + 4
As a reminder, when finding out the equation of a line, we always have two unknowns (here m and c). So we’ll need two equations in these unknowns to find their values, and the equations are to be formed using the conditions given in the problem. Here, the two conditions were (i) the line passing through a point and (ii) forming a triangle of a given area.
I’ll stop here. As we proceed, you’ll realize that a given problem can be solved (in general) using any form of the equation of the line, and some are more efficient than others.