Question #293103

Circles has radius 5 and tangent to the line 3x+4y=24 at the point (2, 412⁄). Illustrate thenfindthe equations of thecircles


1
Expert's answer
2022-02-03T08:53:14-0500

Solution:

r=5r=5

Circle is tangent to the line 3x+4y=24 at (2, 41/2)

Or tangent is 3x+4y=24.

Slope of tangent =34=-\dfrac 34

Let the equation of the circle be (xh)2+(yk)2=r2(x-h)^2+(y-k)^2=r^2

(xh)2+(yk)2=52(xh)2+(yk)2=25 ...(i)\Rightarrow (x-h)^2+(y-k)^2=5^2 \\ \Rightarrow (x-h)^2+(y-k)^2=25\ ...(i)

On differentiating w.r.t xx,

2(xh)+2(yk)y=0y=xhyk     [Slope of tangent]2(x-h)+2(y-k)y'=0 \\\Rightarrow y'=\dfrac{x-h}{y-k}\ \ \ \ \ [Slope\ of\ tangent]

For (2, 41/2), put x=2,y=41/2,y=3/4x=2, y=41/2, y'=-3/4 in above equation.

34=2h412kh=4k883-\dfrac 34=\dfrac{2-h}{\frac {41}2-k} \\ \Rightarrow h=-\dfrac{4k-88}{3}

There is no other info in the problem, as the question is incomplete.

Assume h=0, then k=22

Now put it in (i), we will get the equation of circle.

x2+(y22)2=25x^2+(y-22)^2=25


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