Answered

Get the answers you need at Westonci.ca, where our expert community is always ready to help with accurate information. Get immediate answers to your questions from a wide network of experienced professionals on our Q&A platform. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.

Show that the straight line with equation 2x+y-8 = 0 is a tangent to a circle with equation x^2 + y^2 + 4x - 4y - 12 = 0. What is the equation of the radius drawn to the point of contact?

Sagot :

konrad509
[tex]2x+y-8=0\\ y=-2x+8[/tex]

Now put it into the equation of a circle.

[tex]x^2+(-2x+8)^2+4x-4(-2x+8)-12=0\\ x^2+4x^2-32x+64+4x+8x-32-12=0\\ 5x^2-20x+20=0\\ 5(x^2-4x+4)=0\\ 5(x-2)^2=0\\ (x-2)^2=0\\ x-2=0\\ x=2\\\\ y=-2\cdot2+8=-4+8=4\\\\(x,y)=(2,4)[/tex]

The solution is only one point which means the line is tangent to the circle.
Panoyin
2x + y - 8 = 0
2x - 2x + y - 8 + 8 = 0 - 2x + 8
y = -2x + 8

x² + (-2x + 8)² + 4x - 4(-2x + 8) - 12 = 0
x² + (-2x + 8)(-2x + 8) + 4x - 4(-2x) - 4(8) - 12 = 0
x² + (4x² - 16x - 16x + 64) + 8x - 32 - 12 = 0
x² + 4x² - 32x + 64 + 8x - 32 - 12 = 0
x² + 4x² - 32x + 8x + 64 - 32 - 12 = 0
5x² - 24x - 20 = 0
x = -(-24) +/- √((-24)² - 4(5)(-20))
                         2(5)
x = 24 +/- √(196 + 400)
                   10
x = 24 +/- √(596)
              10
x = 24 +/- 24.413112314674
                        10
x = 2.4 +/- 2.4413112314674
x = 2.4 + 2.4413112314674                                 x = 2.4 - 2.4413112314674
x = 4.8413112314674                                          x = -0.043112314674

y = -2x + 8                                                            y = -2x + 8
y = -2(4.8413112314674) + 8                               y = -2(-0.043112314674) + 8
y = -9.6826224629348 + 8                                   y = 0.086224629348 + 8
y = -1.6836224629348                                         y = 8.086224629348
(x, y) = (4.8413112314674, -1.683622462938)    (x, y) = (-0.043112314674, 8.086224629348)



Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Thank you for trusting Westonci.ca. Don't forget to revisit us for more accurate and insightful answers.