Algebra and graphs：练习题

A parabola has the equation \( y^2 = 12x \). A point \( P \) on the parabola is translated such that the new equation of the curve is \( (y - 3)^2 = 12(x + 1) \). What are the coordinates of the focus of the translated parabola?

A rectangular hyperbola has the equation \( xy = 16 \). The points \( A \) and \( B \) on the hyperbola have coordinates \( (4t_1, \frac{4}{t_1}) \) and \( (4t_2, \frac{4}{t_2}) \) respectively. If the chord \( AB \) passes through the point \( (8, 0) \), find the relationship between \( t_1 \) and \( t_2 \).

The curve \( C \) has the equation \( y = \frac{x^2 + kx + 1}{x} \), where \( k \) is a real constant. It is given that the curve has two distinct stationary points. By considering the discriminant of the quadratic equation in \( x \) formed by the relation \( y = \frac{x^2 + kx + 1}{x} \), find the set of values of \( y \) that are NOT possible for any real value of \( x \).

A curve is defined by the rational function \(y = \frac{x^2 + 4}{x^2 - 9}\). Determine the equations of all its vertical and horizontal asymptotes.

A curve is defined by the rectangular hyperbola equation \( xy = 16 \).

(a) A transformation is applied to this curve consisting of a translation by the vector \( \begin{pmatrix} -2 \\ 3 \end{pmatrix} \). Write down the equation of the transformed curve.

(b) Find the coordinates of the points where the line \( y = x + 5 \) intersects the transformed curve.

(c) Describe a single transformation that maps the original curve \( xy = 16 \) onto the curve \( xy = 4 \).

The curve \( C \) has the equation \( y = \frac{x^2 + 5}{x - 2} \).

(a) Write down the equation of the vertical asymptote to \( C \).

(b) By considering the quadratic equation in \( x \) formed by \( y = \frac{x^2 + 5}{x - 2} \), show that there are no points on the curve for which \( -2 < y < 10 \).

(c) Find the coordinates of the stationary points of the curve \( C \).