(a) If \(f(x) = \frac{2x - 3}{(x^{2} - 1)(x + 2)}\) (i) find the values of x for which f(x) is undefined. (ii) express f(x) in partial fractions. (b) A circ...
Question 1 Report
(a) If \(f(x) = \frac{2x - 3}{(x^{2} - 1)(x + 2)}\)
(i) find the values of x for which f(x) is undefined.
(ii) express f(x) in partial fractions.
(b) A circle with centre (-3, 1) passes through the point (3, 1). Find its equation.
Answer Details
(a)(i) \(f(x)=\dfrac{2x-3}{(x^{2}-1)(x+2)}=\dfrac{2x-3}{(x-1)(x+1)(x+2)}\) is undefined when the denominator is zero:
\[x=1,\quad x=-1,\quad x=-2\]
(ii) Partial fractions. Write
\[\frac{2x-3}{(x-1)(x+1)(x+2)}=\frac{A}{x-1}+\frac{B}{x+1}+\frac{C}{x+2}\]
Cover-up (substitution) gives:
\(x=1:\ A=\dfrac{2(1)-3}{(2)(3)}=-\dfrac{1}{6}\).
\(x=-1:\ B=\dfrac{2(-1)-3}{(-2)(1)}=\dfrac{-5}{-2}=\dfrac{5}{2}\).
\(x=-2:\ C=\dfrac{2(-2)-3}{(-3)(-1)}=\dfrac{-7}{3}\).
\[f(x)=-\frac{1}{6(x-1)}+\frac{5}{2(x+1)}-\frac{7}{3(x+2)}\]
(b) Circle with centre \((-3,1)\) through \((3,1)\): radius \(=\sqrt{(3+3)^{2}+(1-1)^{2}}=6\).
\[(x+3)^{2}+(y-1)^{2}=36\]
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