# Condition For Line To Be Tangent To Circle

Let the line y = mx + c be a tangent to the circle x^{2} + y^{2} = a^{2} at (x1, y1)

The equation of the tangent at (x_{1}, y_{1}) to the circle x^{2} + y^{2} = a^{2} is xx_{1} + yy_{1} = a^{2}

Thus, the equations y = mx + c and xx_{1} + yy_{1} = a^{2} are representing the same straight line and hence their coefficients are proportional.

1/y_{1} = −m/x_{1} = c/a^{2}

x_{1} = −a^{2}m/c

y_{1} = a^{2}/c

(x_{1}, y_{1}) is a point on the circle x^{2} + y^{2} = a^{2}

∴ x_{1}^{2} + y_{1}^{2} = a^{2}

Put the value of x_{1} and y_{1}, you will get the required condition.

**c ^{2} = a^{2}(1 + m^{2})** is the required condition.

The point of contact of the tangent y = mx + c to the circle x^{2} + y^{2} = a^{2} is