If b = a + cp and r = ab + \(\frac{1}{2}\)cp\(^2\), express b\(^2\) in terms of a, c, r.
 

  • A b² = aV + 2cr
  • B b² = ar + 2c²r
  • C b² = a² = \(\frac{1}{2}\) cr²
  • D b² = \(\frac{1}{2}\)ar² + c
  • E b² = 2cr - a²
jamb 1981mathematics

The correct answer is E. b² = 2cr - a²

1. \(b = a + cp\) ...(i)

2. \(r = ab + \frac{1}{2}cp^2\) ...(ii)

Expressing \(b^2\) in terms of \(a\), \(c\), \(r\), we shall first eliminate \(p\) which should not appear in our answer from equation (i):

\(b - a = cp = \frac{b - a}{c}\)

Substitute for \(p\) in equation (ii):

\(r = ab + \frac{1}{2}c\left(\frac{(b - a)^2}{\frac{ab + b^2 - 2ab + a^2}{2c}}\right)\)

Now, simplify:

\(2cr = 2ab + b^2 - 2ab + a^2\)

\(b^2 = 2cr - a^2\)

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