Coordinate systems: แบบฝึกหัด

Find the coordinates of the focus of the parabola with Cartesian equation \(y^2 = 28x\).

A parabola has its focus at the point S(4, 0) and its directrix has the equation x = -4. A point Q lies on the parabola such that the distance from Q to the focus is 13 units. Find the x-coordinate of Q.

The normal to the parabola \(y^2 = 4ax\) at the point \(P(at^2, 2at)\) intersects the x-axis at the point G. Find the coordinates of G.

A rectangular hyperbola H has the Cartesian equation xy = 16. A point P lies on H and is represented by the parameter t = 2 in the standard parametric form \(x = ct\), \(y = \frac{c}{t}\). Find the coordinates of P.

The point P(3, 6) lies on the parabola \(y^2 = 12x\). Find the gradient of the normal to the parabola at P.

A parabola has the Cartesian equation \(y^2 = 20x\). State the coordinates of the focus of this parabola.

The point \(P\) lies on the rectangular hyperbola with equation \(xy = 16\). Given that \(P\) has the parametric form \((4t, \frac{4}{t})\), find the coordinates of \(P\) when \(t = 0.5\).

The normal to the parabola \(y^2 = 4ax\) at the point \(P(at^2, 2at)\) intersects the \(x\)-axis at the point \(Q\). Show that the coordinates of \(Q\) are \((2a + at^2, 0)\) given the normal equation is \(y + tx = 2at + at^3\).

A parabola \(C\) has the Cartesian equation \(y^2 = 12x\).
(a) State the coordinates of the focus \(S\) and the equation of the directrix of \(C\).
(b) The point \(P(3, 6)\) lies on \(C\). Find the equation of the tangent to the parabola at point \(P\), giving your answer in the form \(ax + by + c = 0\).

The rectangular hyperbola \(H\) has the equation \(xy = c^2\), where \(c\) is a positive constant. The point \(P(ct, \frac{c}{t})\), where \(t \neq 0\), lies on \(H\).
(a) Show that the equation of the tangent to \(H\) at \(P\) is \(x + t^2y = 2ct\).
(b) This tangent intersects the x-axis at point \(A\) and the y-axis at point \(B\). Find the coordinates of \(A\) and \(B\) in terms of \(c\) and \(t\).
(c) Show that the area of triangle \(OAB\), where \(O\) is the origin, is independent of \(t\).