Precalculus for Everybody
Lines and Linear Equations
 Using slopes, prove that the \(A = (2, 1)\), \(B = (4, 2)\) and \(C = (1, 1/2)\) are collinear.
In the exercises, from 2 to 9, find an equation of the line that satisfies the given conditions. Write the equation in this form: \(\boldsymbol{y = mx + b}\).
 Pass through (1, 3); has slope 5.
 Pass through the origin; has slope 5.
 Pass through (1, 1) and (2, 3).
 xIntercept 5; yIntercept 2
 Pass through (1, 3), and is parallel to the line \(5y + 3x  6 = 0\).
 Pass through (4, 3), and is perpendicular to the line \(5x + y  2 = 0\).

Is parallel to \(2y + 4x  5 = 0\), and pass through the intersection of the lines:
\[ 5x + y = 4 \quad \text{ and } \quad 2x + 5y  3 = 0 \]
 Pass through (8,6), and intersects the axes at equal distances from the origin.

Given the line \(L:\, 2y  4x  7 = 0\):
 Find the line passing through \(P = (1, 1)\), and perpendicular to \(L\).
 Find the distance from the point \(P = (1, 1)\) to \(L\).
 Using slopes, prove that the points \(A = (3, 1)\), \(B = (6, 0)\) and \(C = (4, 4)\) are the vertices of a right triangle. Find the area of the triangle.

Determine which of the following lines are parallel and which are perpendicular:

\[ L_1:\, 2x + 5y  6 = 0 \]

\[ L_2: \, 4x + 3y  6 = 0 \]

\[ L_3: \, 5x + 2y  8 = 0 \]

\[ L_4: \, 5x + y  3 = 0 \]

\[ L_5: \, 4x + 3y  9 = 0 \]

\[ L_6: \, x + 5y  20 = 0 \]


Find the perpendicular bisector of the segment joining the given points:

\[ (1, \, 0) \, \text{ and } \, (2, \, 3) \]

\[ (1, \, 2) \, \text{ and } \, (3, \, 10) \]

\[ (2, \, 3) \, \text{ and } \,(2, \, 1) \]


The endpoints of one of the diagonals of a rhombus are \((2, 1)\) and \((14, 3)\). Find an equation of the line that contains the other diagonal.
Hint: the diagonals of a rhombus are perpendicular
 Find the distance from the origin to the line \(4x + 3y 15 = 0\).
 Find the distance from the point (0,3) to the line \(5x  12y  10 = 0\).
 Find the distance from the point (1,2) to the line \(x  3y = 5\).
 Find the distance between the parallel lines\(3x  4y = 0\) and \(3x  4y = 10\).
 Find the distance between the parallel lines \(3x  y + 1 = 0\) and \(3x  y + 9 = 0\).
 Find the distance from the point \(Q = (6, 3)\) to the line passing through \(P = (4, 1)\) and parallel to the line \(4x + 3y = 0\).
 Determine the value of \(C\) in the equation of the line \(L\): \(4x +3y + C = 0\). It is known that the distance from the point \(Q = (5, 9)\) to the line \(L\) is 4 times the distance from the point \(P = (3, 3)\) to the line \(L\).
 Find the lines parallel to the line \(5x + 12y  12 = 0\) that are 4 units away from this line.
 Find the equation of the tangent line to the circle \(x^2 + y^2  4x + 6y  12 = 0\) at the point (1, 1).
 Find the equations of the two lines passing through the point \(P = (2, 8)\), and are also tangent to the circle \(x^2 + y^2 = 34\).
 In the above exercise, find the points where the tangent lines make contact with the circle.
 Find the equation of each of the two lines parallel to the line \(2x  2y + 5 = 0\), which are also tangent to the circle \(x^2 + y^2 = 9\).
 Find the equation of the tangent line to the circle \(x^2 + y^2 + 2x + 4y  20 = 0\) at the point (2, 2).
 Find the equation of the circle with center \(C = (1, 1)\), and is also tangent to the line \(5x  12y + 22 = 0\).
 Find the equation of the circle passing through the point $Q = (4, 0)$, and is also tangent to the line \(3x  4y + 20 = 0\) at the point \(P = (12/5, \,16/5)\).
 Find the equation of the circle passing through the points (3, 1) and (1, 3), with center in the line \(3x  y  2 = 0\).
 Both parallel lines, \(2x + y 5 = 0\) and \(2x + y +15 = 0\), are tangent to a circle. One point of tangency is \(B = (2, 1)\). Find an equation of the circle.
 Find an equation of the line passing through the point \(P = (8, 6)\), which also forms a triangle of area 12 with the coordinate axes.

Determine the values of \(k\) and \(n\) in the equations of the lines:
\[ L_1:kx  2y  3 = 0 \quad \text{and} \quad L_2:6x  4y  n = 0, \]\[ \begin{aligned} &L_1: kx  2y  3 = 0 \text{ and} \\[1em] &L_2: 6x  4y  n = 0, \end{aligned} \]
 if \(L_1\) intersects \(L_2\) in a single point.
 if \(L_1\) and \(L_2\) are perpendicular.
 if \(L_1\) and \(L_2\) are parallel and not coincident.
 if \(L_1\) and \(L_2\) are coincident.

Determine for what values of \(k\) and \(n\) the lines:
\[ kx + 8y + n = 0 \quad \text{ and } \quad 2x + ky  1 = 0, \]\[ \begin{aligned} &kx + 8y + n = 0 \text{ and} \\[1em] &2x + ky  1 = 0, \end{aligned} \]
 are parallel and not coincident.
 are coincident.
 are perpendicular.
 The center of a square is \(C = (1, 1)\), and one of its sides is on the line \(x2y = 12\). Find the equations of the lines that containing the other sides.
 Prove that the points \(A = (1, 4)\), \(B = (5, 1)\), \(C = (8, 5)\) and \(D = (4, 8)\) are the vertices of a rhombus (a quadrilateral whose sides have equal length). Verify that the diagonals are perpendicular.

Let \(a\) and \(b\) be the xintersection and the yintersection of a line.
If \(a \neq 0\) and \(b \neq 0\), prove that an equation for this line is \(\frac{x}{a} + \frac{y}{b} = 1\).

Roberto is playing pool in a championship.
He must hit, without spin, the eight ball with the white ball using two sides of the table(as the figure indicates).
If the white ball is on the point \(P = (2, 6)\), and the red ball on \(Q = (3, 2)\), find the points \(A\) and \(B\) of the sides of the table where the ball must make contact to be successful.
If the white ball is on the point \(P = (2, 6)\), and the red ball on \(Q = (3, 2)\), find the points \(A\) and \(B\) of the sides of the table where the ball must make contact to be successful.
Answers

\[ y = 5x – 2 \]

\[ y = 3x \]

\[ y = 2x – 1 \]

\[ y = – \frac{2}{5}x + 2 \]

\[ y = – \frac{3}{5} x + \frac{18}{5} \]

\[ y = \frac{x}{5}+ \frac{11}{5} \]

\[ y = 2x + \frac{41}{23} \]

\[ x + y = 2;\,x – y = 14 \]


\[ y = – \frac{x}{2} + \frac{3}{2} \]

\[ \frac{ 9 \sqrt{5}}{10} \]


\[ 5 \]

\(L_2\) is paralel to \(L_5\); \(L_3\) is perpendicular to \(L_1\); \(L_4\) is perpendicular to \(L_6\).


\[ x – 3y – 6 = 0 \]

\[ x + 2y – 13 = 0 \]

\[ y = 1 \]


\[ y + 3x – 25 = 0 \]

\[ 3 \]

\[ 2 \]

\[ \frac{2}{\sqrt{10}} \]

\[ 2 \]

\[ \frac{ 4 \sqrt{10} }{5} \]

\[ \frac{28}{5} \]

\[ C = 7 \; \text{ or } \; C = \frac{59}{3} \]

\(5x + 12y + 40 = 0\); \(5x + 12y – 64 = 0\)

\[ 3x – 4y + 7 = 0 \]

\(5x – 3y – 34 = 0\); \(3x + 5y + 34 = 0\)

\[ (5, \, 3) \; \text{ y } \; (3, \, 5) \]

\[ x – y – 3 \sqrt{2} = 0; \; x – y + 3 \sqrt{2} = 0 \]\[ \begin{aligned} &x – y – 3 \sqrt{2} = 0; \\[.5em] &x – y + 3 \sqrt{2} = 0 \end{aligned} \]

\[ 3x + 4y – 14 = 0 \]

\[ (x – 1)^2 + (y + 1)^2 = 9 \]

\[ x^2 + y^2 = 16 \]

\[ (x – 2)^2 + (y – 4)^2 = 10 \]

\[ (x + 2)^2 + (y + 1)^2 = 20 \]

\[ 3x – 2y – 12 = 0; \, 3x – 8y + 24 = 0 \]


\[ k \neq 3, \, \text{any } n \]

\[ k = \frac{4}{3}, \, \text{any } n \]

\[ k = 3, \, n \neq 6 \]

\[ k = 3, \, n = 6 \]



\(k = 4\) and \(n \neq 2\) or \(k = 4\) and \(n \neq 2\)

\(k = 4\) and \(n = 2\) or \(k = 4\) and \(n = 2\)

\(k = 0\) and any \(n\)


\(x – 2y – 18 = 0\); \(2x + y + 14 = 0\); \(2x + y – 16 = 0\)

\(A = \left( 0, \, \frac{14}{5} \right)\), \(B = \left( \frac{7}{4}, \, 0\right)\)