Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2014 / Paper 1 / 5

Substituting the line into the hyperbola gives

7 x^{2} - (x + 1)^{2} = 7 ⟺ 3 x^{2} - x - 4 = (3 x - 4) (x + 1) = 0.

The two polynomial roots give all intersections:

(x, y) = (- 1, 0), (4/3, 7/3) .

Both satisfy the original line as well as the curve.

b

Put the equation into standard ellipse form:

\frac{( x - 1 ) ^{2}}{3} + \frac{y ^{2}}{3/2} = 1.

The centre is

(1, 0)

, with horizontal semiaxis

3

and vertical semiaxis

3/2

. Its four axis vertices are

(1 \pm 3, 0)

and

(1, \pm 3/2)

; the ellipse meets the vertical coordinate axis at

(0, \pm 1)

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