Analytic Geometry for the FE Civil Exam

NCEES FE Civil spec: Mathematics and Statistics, A. Analytic geometry. FE Reference Handbook: Mathematics chapter, analytic geometry and conic sections.

Analytic geometry describes lines, circles and other curves with equations, so you can find distances, slopes, intersections and tangents with algebra instead of a drawing. On the FE Civil exam these questions are usually quick points, and the same skills come back in surveying, highway alignment and Mohr's circle.

What the NCEES spec covers

Mathematics section A, "Analytic geometry", covers straight lines (slope, intercepts, parallel and perpendicular lines), distance and midpoint, circles, and the basic conic sections (parabola, ellipse, hyperbola). Expect to rewrite an equation into standard form, read off a center or radius, or find the line tangent to a curve at a point.

Key concepts

Slope is rise over run. Parallel lines share a slope; perpendicular lines have slopes whose product is $- 1$ .
Standard forms make the geometry visible. A circle written as $(x - h)^{2} + (y - k)^{2} = r^{2}$ shows its center $(h, k)$ and radius $r$ directly.
Completing the square converts the general form $x^{2} + y^{2} + D x + E y + F = 0$ into standard form. Divide through first if the squared terms have a coefficient other than 1.
Tangent to a circle is perpendicular to the radius at the point of contact, so its slope is the negative reciprocal of the radius slope.
Conic sections are recognized by their squared terms: one squared variable is a parabola, both with the same sign and different coefficients is an ellipse, opposite signs is a hyperbola.

Core formulas

m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}} y - y_{1} = m (x - x_{1}) y = m x + b

d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} (\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})

(x - h)^{2} + (y - k)^{2} = r^{2} h = - \frac{D}{2}, k = - \frac{E}{2}, r = h^{2} + k^{2} - F

Distance from a point $(x_{0}, y_{0})$ to the line $A x + B y + C = 0$ :

d = \frac{∣ A x _{0} + B y _{0} + C ∣}{A ^{2} + B ^{2}}

Where it is in the FE Reference Handbook

The Mathematics chapter has the straight line, conic section and circle equations, including the general second-degree equation and the standard forms of the parabola, ellipse and hyperbola. Search the PDF for "conic" or "circle" to get there fast.

Worked example

Find the center and radius of the circle $x^{2} + y^{2} - 6 x + 4 y - 12 = 0$ , then find the equation of the line tangent to it at the point $(6, 2)$ .

Step 1: complete the square.

(x^{2} - 6 x + 9) + (y^{2} + 4 y + 4) = 12 + 9 + 4 \Rightarrow (x - 3)^{2} + (y + 2)^{2} = 25

The center is $(3, - 2)$ and the radius is 5. Check the point: $(6 - 3)^{2} + (2 + 2)^{2} = 9 + 16 = 25$ , so it lies on the circle.

Step 2: slope of the radius from the center to the point:

m_{r} = \frac{2 - ( - 2 )}{6 - 3} = \frac{4}{3}

Step 3: tangent slope is the negative reciprocal, $m_{t} = - \frac{3}{4}$ .

y - 2 = - \frac{3}{4} (x - 6) \Rightarrow y = - 0.75 x + 6.5 or 3 x + 4 y - 26 = 0

Check: the distance from the center to this line should equal the radius: $∣3 (3) + 4 (- 2) - 26∣/ 3^{2} + 4^{2} = 25/5 = 5$ . ✓

How it shows up on the exam

Most analytic geometry items take under two minutes if you recognize the form right away. A typical question gives a general second-degree equation and asks for a center, a radius, a vertex or an intercept. Others give two points and ask for the equation of the line through them, the perpendicular bisector, or the distance between them. In civil applications, the same tools locate a point of intersection between two survey lines, find the high point of a parabolic vertical curve, or place the center of Mohr's circle.

A good habit is to sketch the situation, even roughly, before you calculate. A sketch tells you the sign of the slope, which quadrant a center lies in and whether an answer is reasonable. Because the exam is multiple choice, you can also check candidate answers: substitute a given point into each equation and eliminate the choices it does not satisfy. For a tangent line, the distance-to-a-line formula is a fast independent check, as shown in the example below.

Common mistakes

Forgetting to add the completing-the-square constants to both sides of the equation.
Reading the center with the wrong sign: $(x - 3)^{2}$ means h = +3, and $(y + 2)^{2}$ means k = −2.
Not dividing by the leading coefficient when the equation starts with $2 x^{2} + 2 y^{2}$ .
Using the radius slope as the tangent slope instead of its negative reciprocal.
Swapping the order of subtraction in the slope formula between numerator and denominator.

See the analytic geometry example for a distance and midpoint walk-through, and the calculus lesson for tangents to general curves.

Practice Problems

Try these from the free PolyCalc FE Civil problem bank. Each one has a full worked solution.

A line passes through (2, 3) and (6, 11). What is its equation? See the worked solution or practice the Math & Statistics Refresher deck
The straight-line distance between points (1, 2) and (5, 10) is most nearly: See the worked solution or practice the Math & Statistics Refresher deck
Find the center and radius of the circle x² + y² − 6x + 4y − 12 = 0. See the worked solution or practice the Math & Statistics Refresher deck
What is the radius of the circle 2x² + 2y² − 8x + 12y − 6 = 0? See the worked solution or practice the Math & Statistics Refresher deck
Find the equation of the line tangent to the circle x² + y² = 25 at the point (3, 4). See the worked solution or practice the Math & Statistics Refresher deck
The circle (x − 1)² + (y + 2)² = 10 passes through the point (4, −1). Find the tangent line at that point. See the worked solution or practice the Math & Statistics Refresher deck

Related Calculators and Examples

Frequently Asked Questions

How do I find the center and radius of a circle from its general equation?

Group the x and y terms, complete the square for each, and add the same constants to the right side. The result (x - h)^2 + (y - k)^2 = r^2 gives the center (h, k) and radius r. If the squared terms have a coefficient other than 1, divide the whole equation by it first.

How do I find the slope of a line perpendicular to another line?

Take the negative reciprocal of the original slope. A line with slope 4/3 has perpendicular lines with slope -3/4, so the product of the two slopes is -1.

Are conic section formulas given in the FE Reference Handbook?

Yes. The Mathematics chapter lists the straight line, circle, parabola, ellipse and hyperbola equations in standard form, so focus your study on recognizing which form applies and on completing the square quickly.

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