A circle has equation $x^2 + y^2 = 12.25$ - Edexcel - GCSE Maths - Question 24 - 2022 - Paper 2

To verify that point P(2.1, 2.8) lies on the circle, substitute the coordinates into the circle’s equation:

$2.1^2 + 2.8^2 = 12.25$

Calculating this gives:

$4.41 + 7.84 = 12.25$,

which confirms that P lies on the circle.

The slope of the radius from the center (0, 0) to point P(2.1, 2.8) is given by:

$m_{radius} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2.8 - 0}{2.1 - 0} = \frac{2.8}{2.1} \approx 1.333$.

The slope of the tangent line L, being perpendicular to the radius, is the negative reciprocal of the slope of the radius:

$$m_L = -\frac{1}{m_{radius}} = -\frac{1}{1.333} \approx -0.75.$

Using the point-slope form of the equation of a line:

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

substituting in point P(2.1, 2.8) and the slope $m_L$:

$y - 2.8 = -0.75(x - 2.1)$.

Rearranging this gives:

$y = -0.75x + 2.8 + 1.575 = -0.75x + 4.375$.

To express this in the standard form $ax + by = c$, we rearrange:

$0.75x + y = 4.375$ or multiplying through by 8 for integer coefficients:

$$6x + 8y = 35.$