The equation of the common tangent touching the circle (x-3)^2+ y ^2=9 and the parabola y ^2=4 x above the X -axis is

Step-by-step solution

Equation of tangent to parabola. Given y^2 = 4x with a=1 , the general tangent above the X-axis is y = mx + 1 m , with m>0 . Condition for circle tangency. Circle (x-3)^2 + y^2 = 9 has center (3,0) and radius 3 . A line is tangent if the perpendicular distance from center to line equals 3 . For the line in form mx - y + 1 m = 0 , this distance is |3m + 1 m | m^2 + 1 = 3 . Since m>0 and 3m^2+1>0 , the equation simplifies to 3m^2 + 1 m m^2 + 1 = 3 . Squaring both sides and simplifying yields 3m^2 = 1 , so m = 1 3 (ta