Question

a circle S, whise radius is 1 unit, touches X axis at A. The centre Q of S lies in the first quadrant. the tangent from O touhes it at Tand P lies on tis tangentwuch that OAP is a rigth angles triangleat A and its perimemter is 8 units . find length of Qp, equation of circle S. If tangent OT cuts the 2 parallel tangents at O and R then equation of circle curcumscribing triangle ORQ

Answer 1

1. The length QP = 5/3.

2. Equation of circle S: (x – 2)² + (y – 1)² = 1.

3. Equation of the circumcircle of triangle ORQ: (x – 3/4)² + (y – 1)² = (5/4)².

Answer 2

Explanation:

Since S touches the x–axis, its center is (h,1) with A = (h, 0). The tangent from O has length h and slope m determined from the tangency condition giving m = 2h/(h²–1). Using that P = (h, m h) and the perimeter h[1+m+√(1+m²)] = 8 leads to h = 2 and m = 4/3. Then QP = 5/3 and circle S is (x-2)²+(y-1)² = 1. Since the horizontal tangents are y = 0 and y = 2, the line y = 4/3 x meets y = 2 at R = (3/2, 2). The circumcircle of ΔORQ is found to be (x–3/4)²+(y–1)² = (5/4)².