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Find the equation of the tangent lines to x2 + y2-16x -12y +87 =0 at points where y = 4. Give the equations of the lines in the form y = mx + b . Sketch the circle and the tangent lines

 Feb 17, 2022
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Find the equation of the tangent lines to x^2 + y^2-16x -12y +87 =0 at points where y = 4.

 

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\(x^2 + y^2-16x -12y +87 =0\\ (x-16)x+(y-6)y+87=0\\ (x-8)^2+(y-6)^2-13=0\\ (y-6)^2=13-(x-8)^2 \)

\(y=\pm \sqrt{13-(x-8)^2}+6 \\ \large \color{blue}y=-\sqrt{(\sqrt{13})^2-(x-8)^2}+6\\\)

y = 4

\(\sqrt{13-(x-8)^2}+6=4\\ \sqrt{13-(x-8)^2}=-2\\ 13-(x-8)^2=4\\ x-8=\pm \sqrt9\\ \)

x ∈ { 5, 11 }

\(\large \color{blue}P_{f(x)}(5,\ 4)\\\large \color{blue} P_{g(x)}(11,\ 4)\)

 

\(y=- { (13-(x-8)^2})^{0.5}+6\\ \frac{dy}{dx}= -0.5\cdot (13-(x-8)^2)^{-0.5}\cdot 2\cdot (x-8)\\ \large \color{blue}\frac{dy}{dx}= \frac{x-8}{ \pm \sqrt{ 13-(x-8)^2}}\\ \)

 

\(m_{f(5)}=\frac{5-8}{+ \sqrt{ 13-(5-8)^2}} \\ \large \color{blue}m_{f(5)}= -1.5\\ m_{g(11)}= \frac{11-8}{ -\sqrt{ 13-(11-8)^2}}\\ \large \color{blue}m_{g(11)}=1.5\)

 

\(f(x)=m_f(x-x_f)+y_f\\ f(x)=-1.5(x-5)+4\\ \large \color{blue}f(x)=-1.5x+11.5\\ g(x)=m_g(x-x_g)+y_g\\ g(x)=1.5(x-11)+4\\ \large \color{blue}g(x)=1.5x-12.5\)

\(The\ radius\ of\ the\ circle\ is\ \sqrt{3}.\)

 

laugh  !

 Feb 18, 2022
edited by asinus  Feb 18, 2022
edited by asinus  Feb 18, 2022
edited by asinus  Feb 19, 2022
edited by asinus  Feb 19, 2022
edited by asinus  Feb 19, 2022
edited by asinus  Feb 19, 2022
edited by asinus  Feb 19, 2022

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