Tangent condition from discriminant

A question is this type if and only if it requires showing a line is tangent to a conic by setting the discriminant of the intersection equation to zero.

1 questions · Challenging +1.2

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AQA Further Paper 2 2019 June Q8
9 marks Challenging +1.2
8
  1. The line \(y = m x\) is a tangent to \(P _ { 2 }\)
    Prove that \(m = \pm \sqrt { \frac { a } { b } }\)
    Solutions using differentiation will be given no marks.
    8
  2. The line \(y = \sqrt { \frac { a } { b } } x\) meets \(P _ { 2 }\) at the point \(D\).
    The finite region \(R\) is bounded by the \(x\)-axis, \(P _ { 2 }\) and a line through \(D\) perpendicular to the \(x\)-axis. The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid.
    Find, in terms of \(a\) and \(b\), the volume of this solid.
    Fully justify your answer.
  3. Find the eigenvalues and corresponding eigenvectors of the matrix