Question 1

How do you calculate the distance between two points?

Accepted Answer

Use d = √((x₂ - x₁)² + (y₂ - y₁)²), which is the Pythagorean theorem applied to the coordinate differences. Between A(1, 2) and B(5, 4): d = √(4² + 2²) = √20 = 2√5 ≈ 4.4721. The calculator simplifies the radical by pulling perfect squares out of the root (20 = 4 · 5) and shows the decimal next to it.

Question 2

How do you find the equation of the line through two points?

Accepted Answer

Compute the slope m = (y₂ - y₁)/(x₂ - x₁) and use y - y₁ = m(x - x₁), or expand the determinant of x, y, 1 with the two points. For A(1, 2) and B(5, 4): m = 1/2, the slope-intercept form is y = (1/2)x + 3/2 and the standard form is x - 2y + 3 = 0. If x₁ = x₂ the line is vertical and the equation is x = k.

Question 3

What is the difference between the standard, slope-intercept, intercept and parametric forms?

Accepted Answer

The standard form (ax + by + c = 0) works for every line, vertical ones included. The slope-intercept form (y = mx + n) shows the slope m and the y-intercept n, but does not exist for a vertical line. The intercept form (x/p + y/q = 1) shows where the line crosses the axes and does not exist if the line passes through the origin or is parallel to an axis. The parametric form (x = x₀ + t·dₓ, y = y₀ + t·d_y) describes the points with one point and a direction vector.

Question 4

How do you calculate the slope and the angle of inclination?

Accepted Answer

The slope is m = Δy/Δx = tan α, and α lies between 0° and 180°. A positive m gives an acute angle, a negative m an obtuse one, and a vertical line has no m and α = 90°. For m = 1/2, α = arctan(1/2) ≈ 26.5651°, or 26°33′54″. The grade as a percentage, used for ramps and roads, is 100 · m, here 50%.

Question 5

How can you tell whether two lines are parallel or perpendicular?

Accepted Answer

Parallel lines have the same slope, which in standard form means a₁b₂ = a₂b₁. Perpendicular lines have m₁ · m₂ = -1, or a₁a₂ + b₁b₂ = 0, which also covers a vertical line meeting a horizontal one. Example: y = 2x + 1 and x + 2y = 12 have m = 2 and m = -1/2, so they are perpendicular. The Two lines tab reports the relative position and the acute angle between them.

Question 6

How do you find the intersection of two lines?

Accepted Answer

Solve the system made by the two equations. With lines written as ax + by + c = 0 and D = a₁b₂ - a₂b₁, x = (b₁c₂ - b₂c₁)/D and y = (a₂c₁ - a₁c₂)/D. For y = 2x + 1 and x + 2y = 12 the point is (2, 5). If D = 0 the lines do not meet at a single point: they are either distinct parallels or coincident, and the calculator says which.

Question 7

How do you calculate the distance from a point to a line?

Accepted Answer

With the line in standard form, d = |a·x₀ + b·y₀ + c| / √(a² + b²). For P(4, 5) and the line 2x - y + 1 = 0: d = |8 - 5 + 1| / √5 = 4/√5 = 4√5/5 ≈ 1.7889. The same formula, applied to a point on one of two parallel lines, gives the distance between them. The Point and line tab also shows the foot of the perpendicular and the reflected point.

Question 8

How do you check whether three points are collinear and find the triangle area?

Accepted Answer

Compute D = x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂), the determinant of x, y, 1. If D = 0 the points are collinear. Otherwise the triangle area is |D|/2. For (1, 2), (5, 4) and (2, 6): D = 14 and the area is 7. The Triangle tab also shows the sides, perimeter, centroid and the classification by sides and angles.

Question 9

How do you get the equation of a circle and go from general form back to center form?

Accepted Answer

The center form is (x - a)² + (y - b)² = r², with center (a, b) and radius r. Expanding gives the general form x² + y² - 2ax - 2by + (a² + b² - r²) = 0. To go back, complete the squares: x² + y² - 4x + 6y - 12 = 0 becomes (x - 2)² + (y + 3)² = 25, center (2, -3) and radius 5. If the right-hand side comes out zero or negative, there is no real circle.

Question 10

Are the results exact? What are the limits?

Accepted Answer

Yes, when the inputs are numbers or fractions: everything is computed in exact rational arithmetic and radicals are simplified, with the decimal rounded to 4 places beside it. The exceptions are a slope derived from an angle other than 0°, 45°, 90° or 135°, which is flagged as approximate, and the graph, which is only an illustration. Each number accepts up to 18 digits, equations go up to degree 2 (lines and circles) and the calculation is in the plane, not in space.

Analytic Geometry Calculator