A line on a plane

Under what necessary and sufficient condition do the lines $\mathbf{r} = \mathbf{r}_{1} + \mathbf{a}_{1} t$ and $\mathbf{r} = \mathbf{r}_{2} + \mathbf{a}_{2} t\quad \left(\mathbf{a}_{i} \neq 0, i = 1, 2\right)$:

Find the angle between the lines given by their equations:

Two lines are given by the vector equations $(\mathbf{r}, \mathbf{n}) = D$ and $\mathbf{r} = \mathbf{r}_{0} + \mathbf{a} t$, with $(\mathbf{a}, \mathbf{n}) \neq 0$. Find the position vector of the point where the lines meet.

Given a point $M_{0}$ with position vector $\mathbf{r}_{0}$ and a line $(\mathbf{r}, \mathbf{n}) = D$ $(\mathbf{n} \neq 0)$. Find the position vectors of:

Find the distance from point $M_{0}\left(\mathbf{r}_{0}\right)$ to the line given by the equation:

Specify at least one direction vector of the line that:

Find the equation of the line passing through the point $A(- 3, 4)$ and parallel to the line:

Find the equation of the line passing through two given points:

Determine whether the lines of the given pair intersect, are parallel, or coincide; if the lines intersect, find the coordinates of their point of intersection:

For which $a$ do the lines $ax - 4y = 6$ and $x - ay = 3$:

For which $a$ do the three lines $ax + y = 1, \quad x - y = a$, $x + y = a^{2}$ have a common point?

Point $M$ lies on the line $Ax + By + C = 0$; the vector $\overrightarrow {MM_{1}}$ has coordinates $A, B$. Prove that point $M_{1}$ lies in the positive half-plane relative to the line with equation $Ax + By + C = 0$.

Point $M(3, 2)$ is the center of a parallelogram, and its sides lie on some four lines. On each of these lines lies one of the points: $P(2, 1), Q(4, - 1)$, $R(- 2, 0), S(1, 5)$. Find the equations of the lines.

Given two vertices of a triangle $(3, - 1)$ and $(1, 4)$ and the intersection point of its medians $(0, 2)$. Find the coordinates of the third vertex of the triangle and find the equations of its sides.

Find the equation of the line passing through the point $A(1, 2)$ such that the segment of this line contained between the lines $3x + y + 2 = 0$ and $4x + y - 1 = 0$ is bisected at point $A$.

Two medians of a triangle lie on the lines $x + y = 3$ and $2x + 3y = 1$, and the point $A(1, 1)$ is a vertex of the triangle. Find the equations of the sides of the triangle.

Points $K(1, - 2), L(3, 4)$ and $M(5, 0)$ are respectively the midpoints of sides $AD, AB$ and $BC$ of quadrilateral $ABCD$, whose diagonals meet at the point $O(2, 2)$. Find the coordinates of the vertices of the quadrilateral.

Find the equations of the lines passing through the point $A(- 1, 5)$ and equidistant from the two points $B(3, 7)$ and $C(1, - 1)$.

Find the equations of the lines equidistant from the three points $A(3, - 1), B(9, 1)$ and $C(- 5, 5)$.

Through vertex $C$ of parallelogram $ABCD$ a line is drawn meeting the extensions of sides $AB$ and $AD$ respectively at points $K$ and $L$ such that $\left|AK\right| / \left|AB\right| = 5\left|AL\right| / \left|AD\right|$. Find the ratio of the area of the parallelogram to the area of triangle $AKL$.

Specify at least one normal vector of the line that:

Find the equation of the line passing through the point $A(- 3, 4)$ and perpendicular to the line:

Point $A(3, - 2)$ is a vertex of a square, and point $M(1, 1)$ is the intersection point of its diagonals. Find the equations of the sides of the square.

The side length of a rhombus with acute angle $60^{\circ }$ equals 2. The diagonals of the rhombus meet at the point $M(1, 2)$, and the longer diagonal is parallel to the x-axis. Find the equations of the sides of the rhombus.

On the line $5x - y - 4 = 0$ find the point equidistant from the points $A(1, 0)$ and $B(- 2, 1)$.

Find the distance from the point $A(1, - 2)$ to the line given by its equation:

Find the distance between the parallel lines $Ax + By + C_{1} = 0$ and $Ax + By + C_{2} = 0$.

Find the equations of the lines parallel to the line $- 2x + y + 5 = 0$ and at distance $\sqrt{20}$ from the point $A(1, - 2)$.

Point $A$ lies on the line $2x - 3y + 4 = 0$. The distance from point $A$ to the line $3y = 4x$ equals 2. Find the coordinates of point $A$.

Point $A$ lies on the line $x + y = 8$, and $A$ is equidistant from the point $B(2, 8)$ and from the line $x - 3y + 2 = 0$. Find the coordinates of point $A$.

Find the coordinates of all points equidistant from the point $A(- 1, 1)$ and the lines $y = - x$ and $y = x + 1$.

Find the set of points of the plane for which the ratio of the distances to two intersecting lines $A_{1} x + B_{1} y + C_{1} = 0$ and $A_{2} x + B_{2} y + C_{2} = 0$ is a constant $k > 0$.

Given the point $A(1, 2)$ and the line $3x - y + 9 = 0$. Find the coordinates of:

Find the equation of the line symmetric to the line $3x - y + 5 = 0$ with respect to the line $x + y = 1$.

Given the equations of the sides of a triangle: $x + 2y + 1 = 0$, $2x - y - 2 = 0, 2x + y + 2 = 0$. Find the equation of the altitude dropped onto the third side.

Point $H(- 3, 2)$ is the orthocenter of a triangle, two of whose sides lie on the lines $y = 2x$ and $y = - x + 3$. Find the equation of the third side.

Given the coordinates of two vertices of a triangle $A(1, 3)$, $B(2, 5)$ and the orthocenter $H(1, 4)$. Find the coordinates of the third vertex of the triangle and find the equations of its sides.

Point $A(1, 2)$ is the midpoint of one of the bases of a right trapezoid, and point $B(3, - 1)$ is the midpoint of the midsegment. The lateral side perpendicular to the bases lies on the line $\frac{x + 1}{3} = \frac{y - 2}{4}$. Find the equations of the remaining sides of the trapezoid.

Point $A(- 1, 4)$ is a vertex of rhombus $ABCD$, whose diagonals meet at the point $M(2, 3)$. Point $P(3, 1)$ lies on side $AB$. Find the equations of the sides of the rhombus.

Find the equations of the sides of a right triangle, if $C(- 3, 4)$ is the vertex of the right angle, $M(1, 2)$ is the midpoint of the hypotenuse, and point $H(3, 3)$ lies on the hypotenuse.

In triangle $ABC$, points $M_{1}(2, 3), M_{2}(0, 7)$ and $M_{3}(- 2, 5)$ are the midpoints of sides $BC, CA$ and $AB$. Find the equation of line $AB$. Find the angle between the medians $AM_{1}$ and $BM_{2}$.

In parallelogram $ABCD$, vertices $A$ and $C$ have coordinates $(1, 2)$ and $(7, 10)$ respectively, and $H(3, 0)$ is the foot of the altitude dropped from $B$ onto side $AD$. Find the equation of line $AD$. Find the angle between lines $AD$ and $AB$.

In parallelogram $ABCD$, points $K(- 1, 2), L(3, 4)$ and $M(5, 6)$ are the midpoints of sides $AB, BC$ and $CD$ respectively. Find the equation of line $BC$. Find the angle between lines $AL$ and $AM$.

In trapezoid $ABCD$ with bases $AD$ and $BC$, side $CD$ is perpendicular to the bases, points $A$ and $C$ have coordinates $(5, 2)$ and $(- 2, 3)$ respectively, and the extensions of the lateral sides meet at the point $P(- 3, 6)$. Find the equation of line $AD$. Find the angle between lines $AD$ and $AB$.

Points $K(1, 3)$ and $L(- 1, 1)$ are the midpoints of the bases of an isosceles trapezoid, and points $P(3, 0)$ and $Q(- 3, 5)$ lie on its lateral sides. Find the equations of the sides of the trapezoid.

Find the angle between the lines:

Find the equations of the lines passing through the point $A(3, 1)$ and making angles of $45^{\circ }$ with the line $3x = y + 2$.

Point $A(2, 0)$ is a vertex of an equilateral triangle, and the side opposite it lies on the line $x + y - 1 = 0$. Find the equations of the other two sides.

The base of an isosceles triangle lies on the line $x + 2y = 2$, and one of the legs lies on the line $y + 2x = 1$. Find the equation of the other leg of the triangle, given that its distance from the intersection point of the given lines equals $1 / \sqrt{5}$.

Consider the angle between the lines $y = x + 1$ and $y = 7x + 1$ inside which the point $A(1, 3)$ lies. Find the coordinates of the point $B$ lying inside this angle and at distances $4\sqrt{2}$ and $\sqrt{2}$ from the given lines respectively.

Find the equations of the sides of an angle with vertex at the point $B$. A circle of radius $R$ centered at the point $A$ is inscribed in the angle:

Find the equation of the bisector of the angle between the lines $x - 7y = 1$ and $x + y = - 7$ inside which the point $A(1, 1)$ lies.

Find the equation of the bisector of the acute angle between the lines $x - 7y = 1$ and $x + y = - 7$.

Find the equations of the bisectors of the interior angles of the triangle whose sides are given by the equations $3y = 4x$, $4y = 3x, 5x + 12y = 10$.

The vertices of a triangle are the points $A(20, 15)$, $B(- 16, 0), C(- 8, - 6)$. Find the radii and the coordinates of the centers of the inscribed and circumscribed circles.

Given the coordinates of two vertices of a triangle $A(2, - 1)$, $B(1, 5)$ and the point $L(3, 0)$ where its bisectors meet. Find the equations of the sides of the triangle.

Points $A(1, 2)$ and $B(- 3, 0)$ are vertices of isosceles triangle $ABC$, with the base angles $A$ and $B$ equal to $\arccos (1 / \sqrt{5})$. Find the coordinates of vertex $C$, given that it lies on the same side of line $AB$ as the point $M(2, 3)$.

Side $AB$ of triangle $ABC$ is given by the equation $x - y + 1 = 0$, side $BC$ by the equation $2x - 3y + 5 = 0$, side $AC$ by the equation $3x - 4y + 2 = 0$. Find the equation of the line passing through vertex $C$ such that the point where this line meets side $AB$ is at distance $1 / 5$ from side $AC$.

Find the equations of the lines making an angle $\arccos (1 / \sqrt{5})$ with the line $x + 2y - 1 = 0$, and at distance 1 from the point $A(1, 1)$.

Find the radius and the coordinates of the center of the circle passing through the point $A(- 1, 3)$ and tangent to the lines $7x + y = 0$ and $x - y + 8 = 0$.

The hypotenuse of a right triangle lies on the line $2x + y - 2 = 0$, and point $C(3, - 1)$ is the vertex of the right angle. The area of the triangle equals $9 / 4$. Find the equations of the lines on which the legs lie.

Given two coordinate systems $O, \mathbf{e}_{1}, \mathbf{e}_{2}$ and $O^{\prime }, \mathbf{e}_{1}^{\prime }, \mathbf{e}_{2}^{\prime }$. The origin of the second coordinate system has coordinates $a_{10}, a_{20}$ in the first system, and the basis vectors of the second system have coordinates $a_{11}, a_{21}$ and $a_{12}, a_{22}$ respectively in the basis of the first system. In the first coordinate system a line is given by the equation $Ax + By + C = 0$. Find the equation of this line in the second system.

In the plane, three points $A(2, 3), B(1, 4)$, $C(- 1, 2)$ and the line $x - 5y + 7 = 0$ are given. Find the equation of this line in the new coordinate system $A, \overrightarrow {AB}, \overrightarrow {AC}$.

The lines $3y = x + 2$ and $3x + 2y - 5 = 0$ are respectively the axes $O^{\prime } x^{\prime }$ and $O^{\prime } y^{\prime }$ of a new coordinate system, and the point $A(- 1, 2)$ has coordinates $(1, 1)$ in the new system.

In the rectangular coordinate system $O, \mathbf{e}_{1}, \mathbf{e}_{2}$, a line is given by the equation $\sqrt{3} x + 2y - 6 = 0$. The origin of a new rectangular coordinate system is at the point $O^{\prime }(- 2, 3)$, and the basis vectors $\mathbf{e}_{1}^{\prime }$ and $\mathbf{e}_{2}^{\prime }$ are obtained from the vectors $\mathbf{e}_{1}$ and $\mathbf{e}_{2}$ respectively by a rotation through the angle $30^{\circ }$ in the direction of the shortest rotation from $\mathbf{e}_{1}$ to $\mathbf{e}_{2}$. Find the equation of the given line in the coordinate system $O^{\prime }, \mathbf{e}_{1}^{\prime }, \mathbf{e}_{2}^{\prime }$.

Two mutually perpendicular lines, given in a rectangular coordinate system by the equations $2x - y + 1 = 0$ and $x + 2y - 7 = 0$, are respectively the axes $O^{\prime } x^{\prime }$ and $O^{\prime } y^{\prime }$ of a new rectangular coordinate system, and the point $A(2, 0)$ has positive coordinates in the new system.