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Question
I have a question that says:
Find the equation, in standard form, with all integer coefficients, of the line perpendicular to 4x – 2y = 10 and passing through (8, 5).
What do they mean, and do I have to graph multiple slopes?
Answer
Hi Athena,
The question asks us to find an equation for a normal (a line which is perpendicular) to the straight line 4x+2y=10. After we have worked out the equation, we have to put it in standard form, with integer coefficients. (for example, if we consider the equation fro a straight line a*x+b*y+c=0, the question requires that the coefficients "a", "b" and constant "c" must all be integers)
To answer this question, you need to know:
i) how to find the slope (or gradient) of a stright line described by a given equation.
ii) understand that a normal, say, y=nx+c, (which is always at 90 degrees relative to the given equation y=mx+b) has a slope "n" which satisfies the following condition: m*n=-1. This implies that the slope of the normal is related to the slope of the given line by n=-1/m....[#1]
iii) once the slope of the normal has been evaluated, determine the equation of the normal using a point which lies on the normal.
Solution:
i) From 4x+2y=10, we rearrange the equation making y the subject. Intermediate step is 2y=-4x+10, dividing throughout by 2 gives y=-2x+5. Comparing this to y=mx+b, we note that m=-2 and b=5. We interpret "m" as the slope of the given line, and "b" as the y-intercept (where the line crosses the y-axis when x=0).
ii) The gradient of the normal is n=-1/m=1/2. So, we know that the normal has the form y=nx+c, where n=0.5.
iii) Finally, if the normal passes through (8,5), this must satisfy the equation in (ii).
Substituting x=8,y=5 into y=nx+c, (with n=0.5) we get 5=0.5*8+c. Thus, c=1.
The equation of the normal is now fully determined.
Normal equation: y=0.5x+1. ...[#2]
iv) Read the instructions very carefully. We cannot leave the answer in this form, because the question asks for integer coefficients, in standard form.
Multiplying [#2] by 2 on both sides, we readily obtain 2y=x+2. It's now a matter of rearranging this into x-2y+2=0.
Learn this process well, so that you can tackle similar questions.
Cheers
