Problem collection › Analytic geometry
Analytic geometry
12 practice problems for the State Matura in mathematics, with solutions.
Solve the problem in your notebook first, then open the solution. For worked problems, the points for each step are also given.
This topic is not a separate area of the current State Matura exam programme, but the problems are good practice (the 2022 tests had problems like these).
What you need to know
- Rectangular coordinates in the plane: distance between two points, dividing a segment in a given ratio, area of a triangle given by three points
- Equations of a line: general, slope–intercept and intercept form; the line through two points; the line through a point with a given slope; distance from a point to a line
- Relative position of two lines: intersection point, angle between two lines, conditions for parallel and perpendicular lines
Problem 1 Multiple choice
If the distance between the points \((3,y)\) and \((8,7)\) is 13, then \(y\) is equal to
A. 5 or -5
B. 5 or 19
C. 19
D. -5 or 19
Show solution
D. -5 or 19
Problem 2 Multiple choice
If \(( - 2, - 4)\) is the midpoint of \((6, - 7)\) and \((x,y)\) , then the values of \(x\) and \(y\) are:
A. \(x = 2\ ,\ y = 1\)
B. \(x = - 10\ ,\ y = - 1\)
C. \(x = - 10,\ y = 1\)
D. \(x = - 8\ ,\ y = - 1\)
Show solution
B. \(x = - 10\ ,\ y = - 1\)
Problem 3 Multiple choice
The vertices of a triangle are \(A(1,1),\ B(3, - 3)\) and \(C(5, - 3)\). Find the area of the triangle.
A. 3
B. 4
C. 5
D. 6
Show solution
B. 4
\(P=\frac12\cdot\overline{BC}\cdot h=\frac12\cdot2\cdot4=4\)
Problem 4 Short answer
The distance from the point \((3,2)\) to the line \(y = 4x + 2\ \)is ______
Show solution
\[\frac{12}{\sqrt{17}}\]
Problem 5 Worked problem
Find the equation of the line at distance 2 from the origin with slope \(- 2\).
Show solution
Writes the equation of the line in the form \(y = kx + n\) i.e. \(y = -2x + n\) (1 point)
From the point–line distance formula gets \(\left| \frac{n}{\sqrt{5}} \right| = 2\) (1 point)
Problem 6 Worked problem
Find the equation of the set of points equidistant from the points (0,3) and (6,0).
Show solution
Finds the midpoint \((3,\frac{3}{2})\) (1 point)
Finds the slope of the line perpendicular to the line
through (0,3) and (6,0).
\(k = 2\). (1 point)
Writes the equation of the line
\[4x - 2y - 9 = 0\] (1 point)
Problem 7 Multiple choice
The slope of the line \(4x - y + 8 = 0\) is:
A. 2
B. 4
C. 6
D. 8
Show solution
B. 4
Problem 8 Multiple choice
Which of these pairs of equations are parallel lines?
A. \(x - y + 2 = 0\) and \(2x - y + 2 = 0\)
B. \(\ x - y + 1 = 0\) and \(x + y = 0\)
C. \(2x - y + 3 = 0\) and \(2x + y + 3 = 0\)
D. \(2x - y - 1 = 0\) and \(2x - y - 3 = 0\)
Show solution
D. \(2x - y - 1 = 0\) and \(2x - y - 3 = 0\)
Problem 9 Multiple choice
The point \(A(x, - 3)\) lies on the line \(2x - 3y + 1 = 0\). The value of \(x\) is:
A. -5
B. -3
C. -2
D. -1
Show solution
A. -5
Problem 10 Multiple choice
If \(\frac{x}{3} + \frac{y}{- 2} = 1\), then \(y\) is equal to:
A. \(y = - \frac{2}{3}x + 2\)
B. \(y = \frac{2}{3}x - 2\)
C. \(\ y = - \frac{3}{2}x + 2\)
D. \(y = \frac{3}{2}x - 2\)
Show solution
B. \(y = \frac{2}{3}x - 2\)
Problem 11 Multiple choice
The picture shows the line \(4x + 3y = 24\). The length of the segment \(\overline{OC}\) is:
A. 4,8
B. 6,4
C. 8
D. 10
Show solution
A. 4,8
\(A(0,8),\ B(6,0),\ \overline{AB}=10\); \(\overline{OC}=\frac{\overline{OA}\cdot\overline{OB}}{\overline{AB}}=\frac{48}{10}=4{,}8\)
Problem 12 Multiple choice
The lines \(x + y + 5 = 0\) and \(2x + 3y + 10 = 0\) intersect at the point with coordinates:
A. \((5,0)\)
B. \((0,5)\)
C. \(( - 5,0)\)
D. \((0, - 5)\)
Show solution
C. \(( - 5,0)\)