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anonymous

  • 5 years ago

Find the point(s) where the line through the origin with slope 7 intersects the unit circle.

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  1. asnaseer
    • 5 years ago
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    a unit circle has the equation:\[x^2+y^2=1\]a line through the origin with slope 7 would have the equation:\[y=7x\]

  2. asnaseer
    • 5 years ago
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    substitute one into the other and solve for x.

  3. asnaseer
    • 5 years ago
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    do you understand?

  4. asnaseer
    • 5 years ago
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    |dw:1326759365741:dw|

  5. asnaseer
    • 5 years ago
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    do you need more help or can you solve this from here?

  6. anonymous
    • 5 years ago
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    I'm still a little confused

  7. anonymous
    • 5 years ago
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    i understand what to do, but why do y=7x?

  8. asnaseer
    • 5 years ago
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    ok, we know the equation of the straight line is \(y=7x\). so we can substitute this value of y into the first equation to get:\[x^2+(7x)^2=1\]\[x^2+49x^2=1\]\[50x^2=1\]\[x^2=\frac{1}{50}\]\[x=\pm\frac{1}{\sqrt{50}}\]

  9. asnaseer
    • 5 years ago
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    y=7x is the equation of a straight line through the origin with slope 7.

  10. asnaseer
    • 5 years ago
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    so now you have two values for x, substitute each one into the equation \(y=7x\) to get the corresponding value for y and you have then solved this question.

  11. asnaseer
    • 5 years ago
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    does it make sense?

  12. asnaseer
    • 5 years ago
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    the general equation of a straight line is given by:\[y=mx+c\]where 'm' is the slope and 'c' is the y-intercept. In your case, you are told that the slope is 7 (so m=7) and it passes through the origin (so y-intercept is zero, therefore c=0).

  13. anonymous
    • 5 years ago
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    so it would be 7*1/\[\sqrt{50}\] and then 7*-1/\[\sqrt{50}\]

  14. asnaseer
    • 5 years ago
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    correct, so the two values of y will be:\[y=\pm\frac{7}{\sqrt{50}}\]

  15. anonymous
    • 5 years ago
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    what would x equal?

  16. anonymous
    • 5 years ago
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    1?

  17. asnaseer
    • 5 years ago
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    we found x above

  18. asnaseer
    • 5 years ago
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    \[x=\pm\frac{1}{\sqrt{50}}\]

  19. anonymous
    • 5 years ago
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    oh right okay. this is starting to make sense

  20. asnaseer
    • 5 years ago
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    so the two points where the line intersects the circle are:\[(-\frac{1}{\sqrt{50}}, -\frac{7}{\sqrt{50}})\quad\text{and}\quad(\frac{1}{\sqrt{50}}, \frac{7}{\sqrt{50}})\]

  21. anonymous
    • 5 years ago
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    thank you so so much! you are a saint

  22. asnaseer
    • 5 years ago
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    you are very welcome - I'm glad I was able to help :-)

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