O-Level Equations and inequalities: worked solution

Question

The line y=x−4 meets the curve y=2/(x−3) at points A and B. Work out the length of AB.

At a point where the line meets the curve the two y-values are equal, so set the right-hand sides equal: x − 4 = 2/(x − 3) There is a fraction, so multiply both sides by (x − 3) to clear it (this is allowed because x = 3 is not on the curve): (x − 4)(x − 3) = 2 Expand the brackets: x² − 3x − 4x + 12 = 2 x² − 7x + 12 = 2 Move everything to one side so the quadratic equals zero: x² − 7x + 10 = 0 Factorise: two numbers with product +10 and sum −7 are −2 and −5. (x − 2)(x − 5) = 0 x = 2 or x = 5 Find each y-coordinate by substituting back into the simpler of the two equations, the line y = x − 4: x = 2 gives y = 2 − 4 = −2, so A(2, −2) x = 5 gives y = 5 − 4 = 1, so B(5, 1) (Check on the curve: 2/(2 − 3) = −2 and 2/(5 − 3) = 1, both correct.) Now use the distance formula AB = √((x₂ − x₁)² + (y₂ − y₁)²): AB = √((5 − 2)² + (1 − (−2))²) AB = √(3² + 3²) AB = √(9 + 9) AB = √18 AB = 4.24 (3 s.f.)