O-Level Coordinate geometry in two dimensions: worked solution

Question

The line y = −x + 4 touches a circle at A, whose centre is B(−4, −2). Determine the coordinates of A.

Method: a tangent is perpendicular to the radius at the point of contact. So A is the foot of the perpendicular from the centre B onto the line, and we find it by intersecting the line with the perpendicular through B. Step 1 - read the gradient of the tangent. y = −x + 4 is in the form y = mx + c, so its gradient is m = −1. Step 2 - find the gradient of BA. BA is a radius drawn to the point of contact, so BA is perpendicular to the tangent. Perpendicular gradients multiply to −1, so gradient of BA = −1 ÷ (−1) = 1 Step 3 - write the equation of the line BA through B(−4, −2) with gradient 1. Using y − y₁ = m(x − x₁): y − (−2) = 1(x − (−4)) y + 2 = x + 4 y = x + 2 Step 4 - A is where this line meets the tangent, so solve the two equations together. x + 2 = −x + 4 x + x = 4 − 2 2x = 2 x = 1 Step 5 - substitute back to get y. Using y = −x + 4: y = −1 + 4 = 3 (Check with the other line: y = x + 2 = 1 + 2 = 3. Agrees.) So A = (1, 3).