The line L has equation 5y + 12x = 298 A circle, C, has centre (7, 9) L is a tangent to C - AQA - A-Level Maths Pure - Question 6 - 2020 - Paper 2

To find the coordinates of the point of intersection of line L and circle C, we start with the equation of line L:

$5y + 12x = 298$

Rearranging gives:

$y = \frac{298 - 12x}{5}$

Next, we consider the circle C centered at (7, 9) with unknown radius r. The general equation of the circle is:

$(x - 7)^2 + (y - 9)^2 = r^2$

Substituting for y from the line equation into the circle equation:

$(x - 7)^2 + \left(\frac{298 - 12x}{5} - 9\right)^2 = r^2$

This leads to:

$(x - 7)^2 + \left(\frac{298 - 12x - 45}{5}\right)^2 = r^2$

Simplifying:

$(x - 7)^2 + \left(\frac{253 - 12x}{5}\right)^2 = r^2$

Expanding and equating to zero to find the points of intersection:

• Set the discriminant equal to zero to determine tangency:

  $D = b^2 - 4ac = 0$

• Solving for x gives:
  $x = 19$ and subsequently substituting back gives:
  $y = 14$.

Thus, the coordinates of the point of intersection are (19, 14).