Grade 12th passAnalytical Geometry

A circle touches the lines y=x/√3 and y=√3x and has unit radius. Find its equation if center lies in first quadrant.

Pawan joshi

7 Years agoGrade 12th pass

1 Answer

kkbisht

7 Years ago

The two lines y= x/ 3 and y=3 x are touching or tangenst to the given unit circle having radius 1.Therefor using the condition that a line ax+by+c=0 is tangent to a circle if the length of perpendicular from the centre to the line is equal to the radius(=1 ) in this case.

Let (h,k) be the coordinates of centre in the first quadrant means h>0, k>0.

Now length of perpendicular from the centre to the line y= x/ 3 is = (k-h/3)/(1+1/3)=(k-h/3)/(4/3) =1

=>3 k-h=2 -------(1)

and similarly for other line the length of perpendicular from the centre to the line y= 3 x is (k- $\sqrt{}$ 3 h)/ $\sqrt{}$ 4 =1

=> $\sqrt{}$ 3 h -k=2 – ----(2)

(1) =>h= 1+ $\sqrt{}$ 3 and k=1+ $\sqrt{}$ 3 .

Equation of Circle is:

(x-h)² + ( y-k)²=1 => (x-1- $\sqrt{}$ 3)² +(y-1- $\sqrt{}$ 3)² =1

Now just simplify it. You get the required equation of the circle

kkbisht

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A circle touches the lines y=x/√3 and y=√3x and has unit radius. Find its equation if center lies in first quadrant.

Pawan joshi

1 Answer

kkbisht

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