Lines in the Plane

Write the general equation of the line:

The general equation of the line $2 x-3 y-9 = 0$ is given. Write its parametric equation. What segments does this line cut off on the coordinate axes? Find its slope (the coordinate system is rectangular).

The parametric equation of a line is given: $x = 1-3 t, y = 2+2 t$. Write its general equation.

Set up the general and parametric equations of the line passing through the points $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$.

A triangle $A B C: A(-2,3), B(4,1), C(6,-5)$ is given. Write the equation of the line containing the median drawn from the vertex $A$ of this triangle.

A triangle $A B C: A(4,4), B(-6,-1), C(-2,-4)$ is given. Write the equation of the line containing the bisector of the interior angle of the triangle at the vertex $C$. The coordinate system is rectangular.

Through the point $(4,-3)$ draw a line such that the area of the triangle formed by this line and the coordinate axes equals 3. The coordinate system is rectangular.

In a triangle $A B C$, the angles $A$ and $B$ at its base $A B$ are acute, and the lateral sides $B C$ and $A C$ are unequal. Find the locus of the points of intersection of the diagonals of rectangles inscribed in the triangle so that two vertices of the rectangle lie on the base of the given triangle, and the other two on its lateral sides.

Find the locus of the points of intersection of the diagonals of parallelograms inscribed in a given quadrilateral so that the sides of these parallelograms are parallel to the diagonals of the quadrilateral.

Two lines are given, $A x+B y+C = 0$ and $x = x_{0}+a t, y = y_{0}+b t$. Find the necessary and sufficient conditions for these lines to:

Two lines are given

Find the necessary and sufficient conditions for these lines to:

In each of the following cases, determine the relative position of the lines $\ell_{1}$ and $\ell_{2}$. If the lines intersect, find the coordinates of the point of intersection:

Three lines are given, $\ell_{i}: A_{i} x+B_{i} y+C_{i} = 0, i = 1,2,3$. Find the necessary and sufficient conditions for:

Find the necessary and sufficient condition under which three lines, given by the equations $A_{i} x+B_{i} y+C_{i}, i = 1,2$, 3, belong to one pencil (proper or improper).

Three pairwise distinct lines $\ell_{i}: A_{i} x+B_{i} y+C_{i} = 0$, $i = 1,2,3$, belonging to one proper pencil, are given. Let $\ell$ be an arbitrary line not belonging to the same pencil and not parallel to the given lines. In what ratio can the point $R = \ell_{3} \cap \ell$ divide the segment $P Q$, where $P = \ell_{1} \cap \ell , Q = \ell_{2} \cap \ell$?

The equations of two sides of a triangle, $2 x-y = 0,5 x-y = 0$, and the equation $3 x-y = 0$ of one of the medians, are given. Set up the equation of the third side of the triangle, knowing that the point $(3,9)$ lies on it.

Three pairwise distinct lines $\ell_{i}: A x+B y+C_{i} = 0$, $i = 1,2,3$, belonging to one improper pencil, are given. Let $\ell$ be an arbitrary line not belonging to the same pencil. In what ratio can the point $R = \ell_{3} \cap \ell$ divide the segment $P Q$, where $P = \ell_{1} \cap \ell$, $Q = \ell_{2} \cap \ell$?

In each of the following cases, determine the relative position of the three lines:

Let $\ell_{i}(i = 1,2,3)$ be lines in the plane with equations $A_{i} x+B_{i} y+C_{i} = 0$, forming a triangle. Prove that any function of the form $A x+B y+C$ is a linear combination of the left-hand sides of the equations of the lines $\ell_{i}$, i.e., there exist constants $\lambda_{i}$ such that

for all $x, y$.

Describe all triples $\ell_{i}: A_{i} x+B_{i} y+C_{i} = 0, i = 1,2,3$, of lines in the plane possessing the following property: any function of the form $A x+B y+C$ can be represented as a linear combination of the left-hand sides of the equations of the lines $\ell_{i}$, i.e., there exist constants $\lambda_{i}$ such that

for all $x, y$.

Describe all possible families of lines whose equations $A x+B y+C = 0$ can be represented as a linear combination

of the equations $A_{i} x+B_{i} y+C_{i} = 0$ of three fixed lines $\ell_{i}, i = 1,2,3$.

Without finding the point of intersection of the lines $2 x-6 y+3 = 0$ and $5 x + y-2 = 0$, draw through it lines parallel to the coordinate axes, and a line passing through the origin.

Two points $P(1,2)$ and $Q(3,-4)$ are given. Without finding the equation of the line $P Q$, find the point $R$ of its intersection with the line $x+y-2 = 0$ and the ratio $\lambda$ in which it divides the segment $P Q$.

The circle inscribed in a triangle $A B C$ touches its sides $B C, C A$ and $A B$ at the points $P, Q$ and $R$, respectively. Prove that the lines $A P, B Q$ and $C R$ pass through one point.

The sides $B C, C A$ and $A B$ of a triangle $A B C$ are divided by points $P, Q$ and $R$ in the ratios:

Under what necessary and sufficient conditions do

Under what necessary and sufficient condition does the point $(x_{0}, y_{0})$ lie between the two parallel lines $A x+B y+C = 0$ and $A x + B y+D = 0$?

Two parallel lines $3 x-y-2 = 0$ and $3 x-y-7 = 0$ divide the plane into three regions: a strip and two half-planes. In which regions do the following points lie: $A(4,5), B(0,-3), C(5,1), D(1,4)$?

Two points $P\left(x_{1}, y_{1}\right), Q\left(x_{2}, y_{2}\right)$ and a line $\ell : A x+B y + C = 0$ are given. Without finding the point of intersection of the lines $\ell$ and $P Q$, find the ratio $\lambda$ in which it divides the segment $P Q$.

Two points $A(-3,1), B(5,4)$ and the lines

Prove that the line $5 x-y-5 = 0$ intersects the segment of the line $3 x-2 y-6 = 0$ contained between the coordinate axes.

Determine the position of the line $x-7 y+5 = 0$ relative to the triangle whose vertices are at the points $A(3,1), B(-2,4)$ and $C(1,0)$.

Prove that the vector with coordinates $(A, B)$ is always directed toward the positive half-plane relative to the line $A x+B y+C = 0$. Is it true that it is always orthogonal to this line?

Two intersecting lines $A_{1} x+B_{1} y+C_{1} = 0$ and $A_{2} x + B_{2} y+C_{2} = 0$, and a point $(x_{0}, y_{0})$ not lying on them, are given. Find direction vectors $\mathbf{a}_{1}, \mathbf{a}_{2}$ of the sides of the angle between the lines in which the point $\left(x_{0}, y_{0}\right)$ lies.

Find the necessary and sufficient conditions for the point $\left(x_{0}, y_{0}\right)$ to lie inside the triangle formed by the lines $A_{1} x+B_{1} y+C_{1} = 0, A_{2} x+B_{2} y+C_{2} = 0$ and $A_{3} x+B_{3} y+C_{3} = 0$.

A line $x-2 y+1 = 0$, containing the base $B C$ of a triangle $A B C$, and the vertex $A(3,-4)$ opposite this base, are given. Find the equation of the altitude dropped from $A$ to the side $B C$.

Find the projection of the point $(-3,7)$ onto the line $5 x-4 y+2 = 0$.

Find the point symmetric to the point $(0,11)$ with respect to the line $x+7 y-52 = 0$.

On the line $2 x-3 y+18 = 0$ find a point $C$ such that the sum of the distances to the two given points $A(-7,-3)$ and $B(5,5)$ is smallest.

A vertex of a triangle is at the point $(1,3)$, and the bisectors of two of its angles belong to the lines $y+3 = 0$ and $2 x+y+5 = 0$. Write the equation of the side of the triangle opposite this vertex.

Write the equations of the sides of an isosceles trapezoid, knowing the midpoints of its bases $(1,1),(2,8)$ and points $(4,-3),(-15,14)$ on its lateral sides.

The equation of a side of a rhombus, $x+3 y-8 = 0$, and the equation of its diagonal, $2 x+y+4 = 0$, are given. Write the equations of the remaining sides of the rhombus, knowing that the point $(-9,-1)$ lies on the side parallel to the given one.

On the line $2 x-3 y+6 = 0$ find the points at a distance $2 / 5$ from the line $3 x-4 y+11 = 0$.

Set up the equations of the lines parallel to the line $8 x + 6 y-7 = 0$ and situated at a distance 2 from it.

Through the point $(3,-1)$ draw a line at a distance 1 from the point $(2,-3)$.

Through the point of intersection of the lines $3 x+y+10 = 0,4 x+5 y + 6 = 0$, draw a line at a distance 4 from the origin.

Set up the equation of the locus of points equidistant from the lines $7 x+y+10 = 0$ and $x + y+2 = 0$.

Two intersecting lines $A_{1} x+B_{1} y+C_{1} = 0, A_{2} x + B_{2} y+C_{2} = 0$, and a point $\left(x_{0}, y_{0}\right)$ not belonging to either of these lines, are given. Write the equation of the bisector of the angle between the lines in which this point lies.

Write the equation of the bisector of the angle between the lines $2 x-y-1 = 0$ and $11 x+2 y+8 = 0$, inside which the point $(1,2)$ lies.

Write the equation of the bisector of the acute angle between the two lines $x-3 y-1 = 0$ and $3 x-y+5 = 0$.

Write the equations of the tangents to the circle with center $(1,1)$ and radius 2, drawn from the point $(-7,-1)$.

Set up the equations of the bisectors of the interior angles of a triangle whose sides are given by the equations: $3 x-4 y = 0,4 x-3 y = 0,5 x+12 y-10 = 0$.

A vertex $(4,0)$ of a triangle and the altitudes drawn from its two other vertices, $3 x-y-2 = 0, x+y = 0$, are given. Find these vertices.

The center of symmetry of a square is at the point $(-1,0)$, and one of its sides is given by the equation $x+3 y-5 = 0$. Set up the equations of the three other sides.

Set up the equations of the sides of a square, two parallel sides of which pass through the points $(2,1)$ and $(3,5)$, and the other two through the points $(0,1)$ and $(-3,-1)$.

Set up the equation of the circle of radius $\sqrt{5}$, tangent to the line $x+2 y-3 = 0$, knowing that its center lies on the axis $O y$.

Set up the equations of the tangents to the circle

parallel to the line $3 x-4 y = 0$.

Set up the equations of the tangents to the circle

perpendicular to the line $4 x+3 y = 0$.

Find the center of the circle inscribed in the triangle bounded by the coordinate axes and the line $3 x-4 y-5 = 0$.

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

Find the center $C$ and radius $r$ of the circle inscribed in the triangle whose sides are given by the equations: $x+y+12 = 0,7 x+y = 0$, $7 x-y+28 = 0$.

Find the coordinates of the center of the circle:

A triangle with vertices $A(1,1), B(-1,-1), C(7,-5)$ is given. Find the equations of the bisectors of the interior angles, the center $C$ of the inscribed circle, and its radius $r$.

Under the conditions of the preceding problem, find the center $C^{\prime }$ and radius $r^{\prime }$ of the circle tangent to the side $A B$ and to the extensions of the sides $A C$ and $B C$ of the triangle $A B C$.

Problem 321: a triangle with vertices $A(1,1), B(-1,-1), C(7,-5)$ is given.

Given two vertices $A(0,1), B(1,1)$ and the orthocenter $P(-3,4)$ of a triangle, find the third vertex $C$.

(Euler line.) Prove that in every triangle the orthocenter, the centroid, and the center of the circumscribed circle lie on one line.

Write the equations of the sides of a rhombus $A B C D$, knowing the point $M(1,6)$ where its diagonals meet, and points on three of its sides: $P(3,0)$ on $A B, Q(6,6)$ on $B C, R(5,9)$ on $C D$.

Write the equations of the sides of a square $A B C D$, knowing its center $(1,6)$ and points on two of its adjacent sides: $(4,9)$ on $A B,(-5,4)$ on $B C$.

A vertex $P(3,5)$ of an isosceles triangle, the equation of its base $x-2 y+12 = 0$, and the area $S = 15$ are given. Write the equations of the lateral sides.

Let $v_{1}$ and $v_{2}$ be direction vectors of the lines $\ell_{1}$ and $\ell_{2}$, respectively. It is known that the angle from the vector $v_{1}$ to the vector $v_{2}$ equals $6 \pi / 5$. What is the angle:

Determine the (signed) angle from the first line to the second, if their slopes $k_{1} = 1 / 3$ and $k_{2} = -1 / 2$, respectively, are known.

Through the point $(3,1)$ draw lines forming an angle $\pi / 4$ with the line $2 x+3 y-1 = 0$.

Find the interior angles of the triangle formed by the lines

The coordinates $\left(\alpha_{1}, \beta_{1}\right),\left(\alpha_{2}, \beta_{2}\right)$ of direction vectors of two intersecting lines are given. Express in terms of them the angle from the first line to the second.

Two points $A(3,3), B(0,2)$ are given. On the line $x+y-4 = 0$ find a point $M$ from which the segment $A B$ is seen at an angle $\pi / 4$.

The equations of the base of an isosceles triangle, $x+y-1 = 0$, and of one of its lateral sides, $x-2 y-2 = 0$, are given; the point $(-2,0)$ lies on the other lateral side. Find the equation of the third side of the triangle.

Two lines are given: $x+y = 0$ and $x-2 y+6 = 0$. Find a third line such that the second of the given lines is the bisector of the angle between it and the first line.

Two vertices of a triangle $A B C: A(1,2), B(3,4)$ and the tangents of the interior angles at these vertices, $\operatorname {tg} A = -1 / 2, \operatorname {tg} B = 1 / 3$, are given.

Find the third vertex of the triangle, knowing that it lies on the same side of the line $A B$ as the origin.

Two lines $x+y+1 = 0$ and $7 x-y+7 = 0$ are given. Find the cosine of the angle between them in which the point $(1,2)$ lies.

Two lines $A_{1} x+B_{1} y+C_{1} = 0, A_{2} x+B_{2} y+C_{2} = 0$ and a point $\left(x_{0}, y_{0}\right)$ not belonging to either of these lines are given. Find the cosine of the angle $\varphi$ between them in which the given point lies.

Three lines $A_{1} x+B_{1} y+C_{1} = 0, A_{2} x+B_{2} y+C_{2} = 0$ and $A_{3} x + B_{3} y+C_{3} = 0$ form a triangle. Find the cosine of the interior angle of this triangle formed by the first and second lines.

Three lines $A_{1} x+B_{1} y+C_{1} = 0, A_{2} x+B_{2} y+C_{2} = 0$ and $A_{3} x + B_{3} y+C_{3} = 0$ pass through one point. Find the necessary and sufficient condition for the third line to pass through the acute angle formed by the first and second lines.

Find the tangent of the angle $\alpha$ from the axis $O x$ to the line $y = k x+b$, knowing the metric coefficients $g_{11}, g_{12}, g_{22}$ of the basis $\mathbf{e}_{1}, \mathbf{e}_{2}$.

Find the tangent of the angle $\alpha$ from the axis $O x$ to the line $y = k x+b$, if $\left|\mathbf{e}_{1}\right| = \left|\mathbf{e}_{2}\right| = 1,\left(\widehat{\mathbf{e}_{1}, \mathbf{e}_{2}}\right) = \omega$.

Find the tangent of the angle $\varphi$ from the line $y = k_{1} x+b_{1}$ to the line $y = k_{2} x+b_{2}$, knowing the metric coefficients $g_{11}, g_{12}, g_{22}$ of the basis $\mathbf{e}_{1}, \mathbf{e}_{2}$.

Find the tangent of the angle $\varphi$ from the line $y = k_{1} x+b_{1}$ to the line $y = k_{2} x+b_{2}$, if $\left|\mathbf{e}_{1}\right| = \left|\mathbf{e}_{2}\right| = 1,\left(\widehat{\mathbf{e}_{1}, \mathbf{e}_{2}}\right) = \omega$.

Find the necessary and sufficient condition for perpendicularity of the lines $A_{1} x+B_{1} y+C_{1} = 0$ and $A_{2} x+B_{2} y+C_{2} = 0$, knowing the metric coefficients $g_{11}, g_{12}, g_{22}$ of the basis $\mathbf{e}_{1}, \mathbf{e}_{2}$.

Find the necessary and sufficient condition for perpendicularity of two lines $y = k_{1} x+b_{1}$ and $y = k_{2} x+b_{2}$, if $\left|\mathbf{e}_{1}\right| = \left|\mathbf{e}_{2}\right| = 1$, $\left(\widehat{\mathbf{e}_{1}, \mathbf{e}_{2}}\right) = \omega$.

Write the equation of the perpendicular dropped from the point $\left(x_{0}, y_{0}\right)$ onto the line $A x+B y+C = 0$, if $\left|\mathbf{e}_{1}\right| = \left|\mathbf{e}_{2}\right| = 1,\left(\widehat{\mathbf{e}_{1}, \mathbf{e}_{2}}\right) = \omega$.

Knowing the metric coefficients $g_{11}, g_{12}, g_{22}$ of the basis $\mathbf{e}_{1}, \mathbf{e}_{2}$, set up the equations of the family of lines:

Find the distance $d$ from the point $(x_{0}, y_{0})$ to the line $A x+B y + C = 0$, knowing the metric coefficients $g_{11}, g_{12}, g_{22}$ of the basis $\mathbf{e}_{1}, \mathbf{e}_{2}$.

Find the distance $d$ from the point $(x_{0}, y_{0})$ to the line $A x+B y + C = 0$, if $\left|\mathbf{e}_{1}\right| = \left|\mathbf{e}_{2}\right| = 1,\left(\widehat{\mathbf{e}_{1}, \mathbf{e}_{2}}\right) = \omega$.

Find the distance $d$ from the point $(2,1)$ to the line $10 x+56 y-37 = 0$, if $g_{11} = 4, g_{12} = 8, g_{22} = 25$.

Set up the equations of the bisectors of the angles between the coordinate axes, knowing the metric coefficients $g_{11}, g_{12}, g_{22}$ of the basis $\mathbf{e}_{1}, \mathbf{e}_{2}$. Consider the special case $\left|\mathbf{e}_{1}\right| = \left|\mathbf{e}_{2}\right| = 1,\left(\widehat{\mathbf{e}_{1}, \mathbf{e}_{2}}\right) = \omega$.