Science:Math Exam Resources/Courses/MATH110/December 2010/Question 06 (c)

MATH110 December 2010

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Question 06 (c)
A circle of radius 5 centered on the origin has the equation $x^{2}+y^{2}=25$ Prove that the line through the origin and the point (3, -4) is perpendicular to the tangent line in part (b).

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
What is the relationship between slopes of perpendicular lines?

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. If two lines are perpendicular, their slopes are negative reciprocals of each other, that is, if one line has slope $a/b$, the other will have slope $-b/a$. Therefore, what we're really concerned about here is the slope of each line mentioned in the problem statement. We already have the slope of the line from part (b), namely 3/4, so it remains to find the slope of the line through (3, -4) and the origin. This hardly requires any calculus - we just calculate the slope using the two points (0,0) and (3, -4). ${\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}={\frac {-4-0}{3-0}}={\frac {-4}{3}}$ In this case, we have slope $-{\frac {4}{3}}$ and the line from part (b) has slope ${\frac {3}{4}}$. These two numbers are negative reciprocals so the two lines are perpendicular.

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