simultaneous equations

Practise simultaneous equations with an original VectorPaths question, worked explanation and related VCE Methods practice.

4 practice marks

Original practice question

The line $y = ax + 2$ intersects the parabola $y = x^2 + d$ at exactly one point $(3, c)$, where $a$, $c$ and $d$ are real constants. Find the values of $a$, $c$ and $d$.

Worked explanation

• Substitute known point

  \[3a + 2 = 9 + d \implies 3a - d = 7 \quad \text{...(1)}\]

• Single intersection condition

  \[\Delta = a^2 - 4(d-2) = 0 \implies a^2 - 4d + 8 = 0 \quad \text{...(2)}\]

• Solve simultaneously

  \[a^2 - 12a + 36 = 0 \implies (a-6)^2 = 0 \implies a = 6\]

• Find d and c

  \[d = 3(6) - 7 = 11, \quad c = 9 + 11 = 20\]

$a = 6, \; c = 20, \; d = -7$

Original VectorPaths practice; not an official VCAA question or marking rubric.