Example 11. Write the equation of the parabola $y^{2}=5 x$ in polar coordinates
Solution. Substitute the given equation with $x=r \cos \varphi, y=$ $=r \sin \varphi$. For $x \geqslant 0$, i.e., $\varphi \in\left[-\frac{\pi}{2} ; \frac{\pi}{2}\right]$, we get $r^{2} \sin ^{2} \varphi=5 r \cos \varphi$, or $r=\frac{5 \cos \varphi}{\sin ^{2} \varphi}(\varphi \neq 0)$.

## Exercises

1. Form the equation of a circle that is tangent to the $O x$ axis at point $A(6,0)$ and passes through point $B(9,9)$.
2. On the parabola $y^{2}=32 x$, find the point whose distance from the line $4 x+3 y+10=0$ is 2.
3. Find the equation of the set of points equidistant from the circle $x^{2}+4 x+$ $+y^{2}=0$ and the point $M(2,0)$.
4. Form the equation of the set of points whose distance from the point $A(0,1)$ is twice less than the distance to the line $y-4=0$.
5. In the equation of the parabola $y=x^{2}+p x+q$, find the values of the parameters $p$ and $q$, if it is known that the parabola passes through the point $A(-1,3)$ and its axis of symmetry is given by the equation $x=4$.
6. Find the angle between the diagonals of a rectangle, the vertices of which are at the points of intersection of the ellipse $x^{2}+3 y^{2}=12$ and the hyperbola $x^{2}-3 y^{2}=6$. Make a drawing.
7. A hyperbola passes through the point $M\left(6, \frac{3 \sqrt{5}}{2}\right)$, is symmetric with respect to the coordinate axes, and has a real semi-axis $a=4$. Form the equations of the perpendiculars dropped from the left focus of the hyperbola to its asymptotes.
8. An ellipse, symmetric with respect to the coordinate axes, passes through the points $M(2 \sqrt{3}, \sqrt{6})$ and $A(6,0)$. Form its equation, find the eccentricity, and the distance from point $M$ to the foci.
9. Form the equation of the parabola and its directrix, if the parabola passes through the points of intersection of the line $y=x$ and the circle $x^{2}+$ $+y^{2}+6 x=0$ and is symmetric with respect to the $O x$ axis. Draw the line, the circle, and the parabola.
10. Plot the curve by points for the given polar equation for $\varphi \in[0,2 \pi)$ and find the equation of the corresponding curve in the Cartesian coordinate system, whose origin coincides with the pole of the polar system, and the positive x-axis - with the polar axis:
1) $r=\frac{1}{1+\cos \varphi}$; 2) $r=\frac{1}{2+\cos \varphi}$; 3) $r=\frac{1}{3-\cos \varphi}$.
11. Write the canonical equations of the following second-order curves given in polar coordinates:
1) $r=\frac{9}{5-4 \cos \varphi}$
2) $r=\frac{9}{4-5 \cos \varphi}$
3) $r=\frac{3}{1-\cos \varphi}$
12. Form the polar equations of the following curves given by their canonical equations:
1) $\frac{x^{2}}{25}+\frac{y^{2}}{16}=1$ (the polar axis is directed along the $O x$ axis, and the pole is located: a) at the left focus, b) at the right focus);
2) $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ (the polar axis coincides with the $O x$ axis, and the pole - at the center of the hyperbola);
3) $(x-R)^{2}+y^{2}=R^{2}$ (the polar axis coincides with the $O x$ axis, and the pole is located: a) at the center of the circle; b) at the origin).

## Answers

1. $(x-6)^{2}+(y-5)^{2}=25$. 2. $M(0,0), M_{1}(18,-24)$. 3. $x^{2}-\frac{y^{2}}{3}=1$. 4. $\frac{x^{2}}{3}+\frac{y^{2}}{4}=1$. 5. $p=-8 ; q=-6$. 6. $\operatorname{arctg} \frac{3}{4}$. 7. $y= \pm \frac{4}{3}(x+5)$. 8. $\frac{x^{2}}{36}+\frac{y^{2}}{9}=1 ; \varepsilon=\frac{\sqrt{3}}{2}$. 9. $y^{2}=-3 x$. 10. 1) $y^{2}=1-2 x$ - parabola; 2) $\frac{\left(x+\frac{1}{3}\right)^{2}}{\left(\frac{1}{\sqrt{3}}\right)^{2}}+\frac{y^{2}}{\left(\frac{1}{2}\right)^{2}}=1-$ ellipse; 3) $\frac{(x-\sqrt{8})^{2}}{4^{2}}+\frac{y^{2}}{\left(\frac{8 \sqrt{2}}{3}\right)^{2}}=1-$ ellipse. 11. 1) $\frac{x^{2}}{25}+\frac{y^{2}}{9}=1$; 2) $\frac{x^{2}}{16}+\frac{y^{2}}{9}=1$; 3) $y^{2}=6 x$. 12. 1) a) $r=$ $=\frac{16}{5-3 \cos \varphi}$, b) $r=\frac{16}{5+3 \cos \varphi}$; 2) $r^{2}=\frac{b^{2}}{\varepsilon^{2} \cos ^{2} \varphi-1}$; 3) a) $R=R \cos \varphi$, b) $r^{2}-2 r \cos \varphi=R^{2}$.

## § 5. Reduction of the General Equation of a Second-Order Curve to Canonical Form

$1^{\circ}$. Two rectangular coordinate systems $O x y$ and $O_{1} x_{1} y_{1}$ have the following properties (Fig. 2.28): the axes $O x$ and $O_{1} x_{1}$, as well as $O y$ and $O_{1} y_{1}$, are parallel and have the same direction, and the origin $O_{1}$ of the system $O_{1} x_{1} y_{1}$ has known coordinates $O_{1}=O_{1}(a, b)$ relative to the system $O x y$.

Then the coordinates $(x, y)$ and $\left(x_{1}, y_{1}\right)$ of an arbitrary point $M$ of the plane are related by the equations

$$
\left\{\begin{array} { l } 
{ x = x _ { 1 } + a , } \\
{ y = y _ { 1 } + b ; }
\end{array} \quad \left\{\begin{array}{l}
x_{1}=x-a \\
y_{1}=y-b
\end{array}\right.\right.
$$

The formulas (3) are called the formulas for the transformation of coordinates under a parallel translation of the coordinate axes.

$2^{\circ}$. Suppose that the rectangular coordinate systems $O x y$ and $O x_{1} y_{1}$ have a common origin, and the axis $O x_{1}$ forms an angle $\alpha$ with the axis $O x$ (under $\alpha$ we mean the angle of rotation of the axis $O x_{1}$ relative to $O x$). Then

![](https://cdn.mathpix.com/cropped/2024_05_22_8f5be56c3f01f571fd60g-032.jpg?height=275&width=327&top_left_y=189&top_left_x=202)

Fig. 2.28

![](https://cdn.mathpix.com/cropped/2024_05_22_8f5be56c3f01f571fd60g-032.jpg?height=284&width=