C is a circle with centre the origin - Edexcel - GCSE Maths - Question 1 - 2020 - Paper 2

To find the angle (θ) that the tangent makes with the x-axis, we use the tangent function:

$\tan(θ) = m$

Substituting the value of m:

$θ = \tan^{-1}\left(\frac{1}{2}\right)$

Using the gradient-point form of the equation of a line, we can express the equation of the tangent line as:

$y - y_1 = m(x - x_1)$

Using the point (0, 10):

$y - 10 = \frac{1}{2}(x - 0)$

This can be simplified to:

$y = \frac{1}{2}x + 10$

The equation of a circle with center at the origin (0,0) is given by:

$x^2 + y^2 = r^2$

To find the radius (r), we can use the distance from the center (0,0) to a point on the tangent which is at (0, 10):

$r = \sqrt{(0 - 0)^2 + (10 - 0)^2} = \sqrt{100} = 10$

Hence, the equation of the circle is:

$x^2 + y^2 = 10^2$

This simplifies to:

$x^2 + y^2 = 100$