Answer to Question #276120 in Analytic Geometry for Nick

Question #276120

what is the equation of a circle passing through (6,2) and tangent to the line x-4y-15=0 at (3,-3)

1
Expert's answer
2021-12-16T16:11:55-0500

The equation of the circle with center (a,b)(a, b) and radius rr is



(xa)2+(yb)2=r2(x-a)^2+(y-b)^2=r^2

Differentiate both sides with respect to xx



2(xa)+2(yb)y=02(x-a)+2(y-b)y'=0

Solve for yy'



y=xayby'=-\dfrac{x-a}{y-b}

The slope of the line tangent to the circle at (3,3)(3, -3)



slope=m=3a3b=3a3+bslope=m=-\dfrac{3-a}{-3-b}=\dfrac{3-a}{3+b}

The equation of the tangent line tangent to the circle at (3,3)(3, -3) is



y(3)=3a3+b(x3)y-(-3)=\dfrac{3-a}{3+b}(x-3)y=3a3+bx33a3+b3y=\dfrac{3-a}{3+b}x-3\cdot\dfrac{3-a}{3+b}-3y=3a3+bx36+ba3+by=\dfrac{3-a}{3+b}x-3\cdot\dfrac{6+b-a}{3+b}

The line x4y15=0x-4y-15=0



y=14x154y=\dfrac{1}{4}x-\dfrac{15}{4}

Then



{3a3+b=1436+ba3+b=154\begin{cases} \dfrac{3-a}{3+b}=\dfrac{1}{4}\\ \\ -3\cdot\dfrac{6+b-a}{3+b}=-\dfrac{15}{4} \end{cases}{b=94a6+ba3a=5\begin{cases} b=9-4a\\ \\ \dfrac{6+b-a}{3-a}=5 \end{cases}b=94a,aRb=9-4a, a\in \R


Substitute



(xa)2+(y9+4a)2=r2(x-a)^2+(y-9+4a)^2=r^2


Point (3,3)(3,-3)



(3a)2+(39+4a)2=r2(3-a)^2+(-3-9+4a)^2=r^296a+a2+14496a+16a2=r29-6a+a^2+144-96a+16a^2=r^2r2=17a2102a+153r^2=17a^2-102a+153

Point (6,2)(6,2)



(6a)2+(29+4a)2=r2(6-a)^2+(2-9+4a)^2=r^23612a+a2+4956a+16a2=r236-12a+a^2+49-56a+16a^2=r^2r2=17a268a+85r^2=17a^2-68a+85r2=17a2102a+153=17a268a+85r^2=17a^2-102a+153=17a^2-68a+8534a=6834a=68a=2a=2b=94(2)=1b=9-4(2)=1r2=17(2)268(2)+85=17r^2=17(2)^2-68(2)+85=17

The equation of the circle is



(x2)2+(y1)2=17(x-2)^2+(y-1)^2=17

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