from the differential equation of a family of circles touching y axis at origin
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see the attachment,
here it is clear that ,
centre of circle is (r,0)
hence, equation of circle is (x -r)² + y² = r² or, x² + y² = 2xr
2r = (x² + y²)/x _________________(1)
now differentiate equation of circle
2x + 2y.dy/dx = 2r
put equation (1)
2x + 2y.dy/dx = (x² + y²)/x
2y.dy/dx = (x² + y²)/x - 2x = (y² - x²)/x
dy/dx = (y² - x²)/2xy
answer
here it is clear that ,
centre of circle is (r,0)
hence, equation of circle is (x -r)² + y² = r² or, x² + y² = 2xr
2r = (x² + y²)/x _________________(1)
now differentiate equation of circle
2x + 2y.dy/dx = 2r
put equation (1)
2x + 2y.dy/dx = (x² + y²)/x
2y.dy/dx = (x² + y²)/x - 2x = (y² - x²)/x
dy/dx = (y² - x²)/2xy
answer
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