If the line y=√x+k touches the circle x2+y2=16, then find the value of k.
Consider the given equation of a circle as x2+y2=16
Centre is (0, 0) and radius =4 as shown figure.
AB be a line passing through centre of circle.
Tangent y=√x+k touches the circle at B(a,b)
AB is perpendicular to tangent.
Slope of AB=−1√
Equation of AB is
y=−1√× [AB passes through centre (0, 0)]
Substituting (2) in (1), we get,
B(a,b) is on y=(2√)+k±2=±6+kk=±8