Example 5. The polar equation $r=\frac{7}{2 \cos \varphi-5 \sin \varphi}$ can be written in rectangular coordinates.
Solution. Let's rewrite the given equation as $2 r \cos \varphi - 5 r \sin \varphi = 7$ and use the formulas: $r \cos \varphi = x, r \sin \varphi = y$. We obtain the equation of the line: $2 x - 5 y = 7$.

## Exercises

1. Form the equation of the line passing through the point of intersection of the medians of a triangle with vertices $A(2,5), B(5,-1)$, and $C(8,3)$, perpendicular to the line $x + y + 4 = 0$.
2. Find the equations of the sides of a triangle if the coordinates of their midpoints are $(2,1), (4,3)$, and $(-2,5)$.
3. Form the equations of the sides of a triangle if the coordinates of one of its vertices are $(1,3)$ and the equations of two medians are $x - 2 y + 1 = 0$ and $y - 1 = 0$.
4. Form the equations of three sides of a square if it is known that the fourth side is a segment of the line $4 x + 3 y - 12 = 0$, the ends of which lie on the coordinate axes.
5. Find the distance from the point $M(2,-1)$ to the line that intercepts segments $a=8, b=6$ on the positive directions of the $O x$ and $O y$ axes, respectively.
6. Given the coordinates of the vertices of triangle $ABC: A(-2,-4)$, $B(2,-1), C(-4,1)$.
1) Form the equations of the sides of the triangle, the median $AM$, and the altitude $BH$.
2) Describe the system of inequalities for the region belonging to triangle $ABC$.
3) Find the internal angle $B$ of triangle $ABC$.
4) Find the coordinates of points $M$ and $H$. Make a drawing.
7. Given the coordinates of three vertices of parallelogram $ABCD: A(4,2)$, $B(-1,-2), C(-3,2)$.
1) Find the coordinates of the fourth vertex $D$ and the point of intersection of the diagonals of the parallelogram.
2) Calculate the area of the parallelogram and the lengths of its heights.
3) Verify the equality $A C^{2} + B D^{2} = 2(A B^{2} + A D^{2})$. Make a drawing.
8. Given the equation of one side of a rhombus $x - 3 y + 10 = 0$ and one of its diagonals $x + 4 y - 4 = 0$. The diagonals of the rhombus intersect at point $P(0,1)$. Find the equations of the other sides and diagonals of the rhombus and the coordinates of its vertices. Make a drawing.
9. Given the equations of the sides of a triangle: $y = x + 2, y = -\frac{1}{2} x + 1$ and the point $D(4,2)$ of intersection of its medians. Write the equation of the third side.
10. Write the equations of the sides of a square, two opposite vertices of which are points $A(1,3), C(6,-2)$.
11. Form the equations of all sides of a square if it is known that the coordinates of two adjacent vertices of the square are $A(1,4)$ and $B(4,0)$, and vertex $C$ is on the y-axis.
12. Determine the angle between the line $2 x - 3 y + 5 = 0$ and the line $x = 0$.
13. Given a triangle with vertices $A(4,6), B(-3,0), C(2,3)$. Find the equation of the bisector $AD$.

## Answers

1. $3 x - 3 y - 8 = 0$. 2. $x + 3 y - 5 = 0, x - y + 7 = 0, x + y - 7 = 0$. 3. $x + 2 y - 7 = 0, x - y + 2 = 0, x - 4 y - 1 = 0$. 4. $3 x - 4 y - 9 = 0$, $3 x - 4 y + 16 = 0, 4 x + 3 y - 37 = 0$ (or $4 x + 3 y + 13 = 0$). 5. 4.4. 6. 1) $3 x - 4 y - 10 = 0, 5 x + 2 y - 18 = 0, x + 3 y + 1 = 0$; 2) $3 x - 4 y - 10 \leqslant 0, 5 x + 2 y + 18 \geqslant 0, x + 3 y + 1 \leqslant 0$; 3) $B = \operatorname{arctg} \frac{13}{9}$; 4) $M(-1,0), H(-72 / 29, -82 / 29)$. 7. 1) $D(2,6), O(1 / 2, 2)$; 2) 28, $14 / \sqrt{5}, 28 / \sqrt{41}$. 8. $39 x + 37 y + 82 = 0, 39 x + 37 y - 156 = 0$, $x + 4 y - 4 = 0, x - 3 y - 4 = 0, 5 x - 7 y - 28 = 0$; $A(-4,2)$, $B(7 / 11, 39 / 11)$, $C(4,0)$, $D(-7 / 11, -17 / 11)$. 9. $5 x + y - 34 = 0$. 10. $x = 1, x = 6, y = -2, y = 3$. 11. $4 x + 3 y - 16 = 0, 3 x - 4 y - 12 = 0$, $4 x + 3 y + 9 = 0, 3 x - 4 y + 13 = 0$. 12. $\operatorname{arctg} \frac{3}{2}$. 13. $5 x - 3 y - 2 = 0$.

## § 4. Second-Order Curves

The second-order curves include the following four lines: circle, ellipse, hyperbola, and parabola. The coordinates $x, y$ of the points of each of these lines satisfy the corresponding second-degree equation in terms of variables $x$ and $y$.

Below, by the geometric locus of points (abbreviated as GMP), we mean a certain set of points on the plane, the coordinates of which satisfy a certain condition. We will define the second-order curves through the GMP, indicating the properties of these points.

## 4.1. Circle

A circle of radius $R$ with center at point $M_{0}(x_{0}, y_{0})$ is the GMP of points equidistant from point $M_{0}$ at a distance $R$.

The canonical equation of a circle is given by $\left(x - x_{0}\right)^{2} + \left(y - y_{0}\right)^{2} = R^{2}$

## Examples with Solutions