Mathematics · Chapter 3
Study notes aligned to the official NEB syllabus.
This chapter extends coordinate geometry in two directions. In the plane it completes the study of the conic sections with the standard equations of the ellipse and the hyperbola. It then moves into three dimensions, describing the orientation of a line by its direction cosines and ratios and studying the plane in its several standard forms. The chapter closes with the vector product of two vectors, the algebraic tool that computes areas and perpendiculars in space and that connects naturally to the geometry of planes.
An ellipse is the locus of a point the sum of whose distances from two fixed points (the foci) is constant. With the foci on the $x$-axis and centre at the origin, its standard equation is
$$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \qquad a > b > 0.$$
Here $2a$ is the length of the major axis (along the $x$-axis) and $2b$ the length of the minor axis. The foci are at $(\pm c, 0)$ where
$$c^2 = a^2 - b^2,$$
and the eccentricity is $e = \dfrac{c}{a} < 1$. The vertices are $(\pm a, 0)$, the directrices are the lines $x = \pm\dfrac{a}{e}$, and the latus rectum (the chord through a focus perpendicular to the major axis) has length $\dfrac{2b^2}{a}$.
Worked example. For the ellipse $\dfrac{x^2}{25} + \dfrac{y^2}{9} = 1$, we read $a^2 = 25$, $b^2 = 9$, so $a=5$, $b=3$ and
$$ \begin{aligned} c &= \sqrt{25 - 9} \ &= 4 \ \qquad e &= \frac{4}{5}. \end{aligned} $$
The foci are $(\pm 4, 0)$, the vertices $(\pm 5, 0)$, and the latus rectum has length $\dfrac{2\cdot 9}{5} = \dfrac{18}{5}$.
A hyperbola is the locus of a point the difference of whose distances from two fixed foci is constant. Its standard equation, with foci on the $x$-axis, is
$$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1.$$
The transverse axis has length $2a$ and the conjugate axis length $2b$. The foci are at $(\pm c, 0)$ where
$$c^2 = a^2 + b^2,$$
and the eccentricity is $e = \dfrac{c}{a} > 1$. The vertices are $(\pm a, 0)$, the latus rectum has length $\dfrac{2b^2}{a}$, and the two asymptotes are the lines $y = \pm\dfrac{b}{a}x$, which the curve approaches at infinity.
Worked example. For $\dfrac{x^2}{16} - \dfrac{y^2}{9} = 1$ we have $a=4$, $b=3$, so
$$ \begin{aligned} c &= \sqrt{16 + 9} \ &= 5 \ \qquad e &= \frac{5}{4}. \end{aligned} $$
The foci are $(\pm 5, 0)$ and the asymptotes are $y = \pm\tfrac34 x$.
This chapter extends coordinate geometry in two directions. In the plane it completes the study of the conic sections with the standard equations of the ellipse and the hyperbola. It then moves into three dimensions, describing the orientation of a line by its direction cosines and ratios and studying the plane in its several standard forms. The chapter closes with the vector product of two vectors, the algebraic tool that computes areas and perpendiculars in space and that connects naturally to the geometry of planes.
An ellipse is the locus of a point the sum of whose distances from two fixed points (the foci) is constant. With the foci on the -axis and centre at the origin, its standard equation is
Here is the length of the major axis (along the -axis) and the length of the minor axis. The foci are at where
and the eccentricity is . The vertices are , the directrices are the lines , and the latus rectum (the chord through a focus perpendicular to the major axis) has length .
Worked example. For the ellipse , we read , , so , and
The foci are , the vertices , and the latus rectum has length .
A hyperbola is the locus of a point the difference of whose distances from two fixed foci is constant. Its standard equation, with foci on the -axis, is
The transverse axis has length and the conjugate axis length . The foci are at where
and the eccentricity is . The vertices are , the latus rectum has length , and the two asymptotes are the lines , which the curve approaches at infinity.
Worked example. For we have , , so
The foci are and the asymptotes are .