Coordinate Geometry

The One Thing

Coordinate geometry represents geometric objects algebraically, allowing points, straight lines, and circles to be analysed through coordinates, equations, distances, slopes, angles, intersections, and tangency conditions. The central technique is to select the equation or formula that matches the information given.

Definitions and Results

• Cartesian Coordinate System: A plane formed by two perpendicular axes, the x-axis and y-axis, used to locate points as ordered pairs $(x,y)$.
• Coordinates of a Point: In $(x,y)$, $x$ is the abscissa or x-coordinate and $y$ is the ordinate or y-coordinate.
• Distance Formula: The distance between $P(x_1,y_1)$ and $Q(x_2,y_2)$ is

$$\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}.$$

• Section Formula: If a point divides the line joining $(x_1,y_1)$ and $(x_2,y_2)$ internally in the ratio $m:n$, its coordinates are

$$\left(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n}\right).$$

• Midpoint Formula: The midpoint of $(x_1,y_1)$ and $(x_2,y_2)$ is

$$\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right).$$

• Slope or Gradient: The slope of a non-vertical line through $(x_1,y_1)$ and $(x_2,y_2)$ is

$$m=\frac{y_2-y_1}{x_2-x_1}.$$ It measures the line’s inclination to the positive x-axis. A vertical line has undefined slope and equation $x=a$; a horizontal line has slope $0$ and equation $y=b$.

• Angle of Inclination: If a line makes an angle $\theta$ with the positive x-axis, then

$$m=\tan \theta ,0^{\circ }\le \theta <180^{\circ }.$$

• Equation of a Straight Line: An equation in $x$ and $y$ satisfied by every point on a particular line.
• Slope-Intercept Form:

$$y=mx+c,$$ where $m$ is the slope and $c$ is the y-intercept.

• Point-Slope Form: The line with slope $m$ through $(x_1,y_1)$ is

$$y-y_1=m(x-x_1).$$

• Two-Point Form: The line through $(x_1,y_1)$ and $(x_2,y_2)$ is

$$y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1),$$ provided $x_1\ne x_2$. Equivalently, $$\frac{y-y_1}{y_2-y_1}=\frac{x-x_1}{x_2-x_1},$$ when the denominators are non-zero.

• Intercept Form: If a line cuts the axes at $(a,0)$ and $(0,b)$, its equation is

$$\frac{x}{a}+\frac{y}{b}=1.$$

• General Form of a Line:

$$Ax+By+C=0,$$ where $A$ and $B$ are not both zero.

• Parallel Lines: Two non-vertical lines are parallel when their slopes are equal. The lines

$$Ax+By+C_1=0,Ax+By+C_2=0$$ are distinct and parallel if $C_1\ne C_2$.

• Perpendicular Lines: Lines with slopes $m_1$ and $m_2$ are perpendicular when

$$m_1{m}_2=-1,$$ provided both slopes are finite.

• Angle Between Two Lines: For slopes $m_1$ and $m_2$,

$$\tan \theta =\left|\frac{m_2-m_1}{1+m_1{m}_2}\right|,$$ where $\theta$ is the acute angle between the lines.

• Distance of a Point from a Line: The perpendicular distance from $(x_1,y_1)$ to $Ax+By+C=0$ is

$$\frac{|A{x}_1+B{y}_1+C|}{\sqrt{A^2+B^2}}.$$

• Distance Between Parallel Lines: The distance between

$$Ax+By+C_1=0\text{and}Ax+By+C_2=0$$ is $$\frac{|C_1-C_2|}{\sqrt{A^2+B^2}}.$$

• Circle: A circle is the locus of a point that remains at a constant distance, called the radius, from a fixed point called the centre.
• Standard Equation of a Circle: A circle with centre $(h,k)$ and radius $r$ has equation

$$(x-h{)}^2+(y-k{)}^2=r^2.$$

• General Equation of a Circle:

$$x^2+y^2+2gx+2fy+c=0$$ represents a circle with centre $$(-g,-f)$$ and radius $$\sqrt{g^2+f^2-c},$$ when the radius is real and positive.

• Circle with Centre at the Origin: A circle of radius $r$ centred at $(0,0)$ has equation

$$x^2+y^2=r^2.$$

• Locus: The path or set of all points satisfying a specified geometric condition.
• Tangent to a Circle: A tangent is a line that touches a circle at exactly one point. The radius drawn to the point of contact is perpendicular to the tangent.
• Condition for a Line to Touch a Circle: A line is tangent to a circle when the perpendicular distance from the centre to the line equals the radius.
• Circle Radius Conditions: For

$$x^2+y^2+2gx+2fy+c=0,$$ a real circle requires $$g^2+f^2-c>0.$$ If $g^2+f^2-c=0$, the circle has zero radius and is called a point circle.

• Axis Intercepts of a Circle: A circle with centre $(h,k)$ and radius $r$ cuts the x-axis at points found by putting $y=0$, and cuts the y-axis at points found by putting $x=0$.
• Circle with a Given Diameter: The circle with diameter endpoints $(x_1,y_1)$ and $(x_2,y_2)$ is

$$(x-x_1)(x-x_2)+(y-y_1)(y-y_2)=0.$$

• Circle Through Three Points: To find the equation of a circle through three non-collinear points, substitute the points into

$$x^2+y^2+2gx+2fy+c=0$$ and solve for $g$, $f$, and $c$.

• Tangent at a Point on an Origin-Centred Circle: The tangent at $(x_1,y_1)$ to

$$x^2+y^2=r^2$$ is $$x{x}_1+y{y}_1=r^2,$$ provided $(x_1,y_1)$ lies on the circle.

• Line-Circle Intersections: Zero, one, or two intersection points correspond respectively to no intersection, tangency, or a secant line.

Worked Methods

1. Finding the distance between two points

• Identify the coordinates $P(x_1,y_1)$ and $Q(x_2,y_2)$.
• Find the horizontal and vertical differences:

$$x_2-x_1,y_2-y_1.$$

• Substitute into

$$PQ=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}.$$

2. Finding a midpoint

• Add the x-coordinates and divide by $2$.
• Add the y-coordinates and divide by $2$.
• Use

$$\text{midpoint}=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right).$$

3. Applying the section formula

• Identify the endpoints $(x_1,y_1)$ and $(x_2,y_2)$.
• Identify the internal division ratio $m:n$.
• Substitute into

$$\left(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n}\right).$$

4. Finding the slope of a line

• Identify two points $(x_1,y_1)$ and $(x_2,y_2)$.
• Calculate

$$m=\frac{y_2-y_1}{x_2-x_1}.$$

• If $x_2-x_1=0$, the line is vertical, has undefined slope, and is written as $x=a$.
• If $y_2-y_1=0$, the line is horizontal, has slope $0$, and is written as $y=b$.

5. Finding the equation of a line from a point and slope

• Identify the known point $(x_1,y_1)$ and slope $m$.
• Use point-slope form:

$$y-y_1=m(x-x_1).$$

• Expand or rearrange if a different form is required.

6. Finding the equation of a line from two points

• Identify $(x_1,y_1)$ and $(x_2,y_2)$.
• Calculate the slope:

$$m=\frac{y_2-y_1}{x_2-x_1}.$$

• Substitute into

$$y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1),$$ provided $x_1\ne x_2$.

7. Finding the equation of a line from intercepts

• Identify the x-intercept $(a,0)$ and y-intercept $(0,b)$.
• Substitute $a$ and $b$ into

$$\frac{x}{a}+\frac{y}{b}=1.$$

For a line in general form $Ax+By+C=0$:

• Put $y=0$ to find the x-intercept.
• Put $x=0$ to find the y-intercept.

8. Testing lines for parallelism or perpendicularity

• Obtain the slopes $m_1$ and $m_2$.
• For parallel lines, test whether

$$m_1=m_2.$$

• For perpendicular lines, test whether

$$m_1{m}_2=-1,$$ provided both slopes are finite.

• If the lines are given in the form

$$Ax+By+C_1=0,Ax+By+C_2=0,$$ they are distinct and parallel when $C_1\ne C_2$.

9. Finding the angle between two lines

• Determine the slopes $m_1$ and $m_2$.
• Substitute them into

$$\tan \theta =\left|\frac{m_2-m_1}{1+m_1{m}_2}\right|.$$

• Interpret $\theta$ as the acute angle between the lines.

10. Finding the distance from a point to a line

• Write the line in the form

$$Ax+By+C=0.$$

• Substitute the point $(x_1,y_1)$.
• Use

$$d=\frac{|A{x}_1+B{y}_1+C|}{\sqrt{A^2+B^2}}.$$

11. Finding the distance between parallel lines

• Ensure the lines have the same $A$ and $B$ coefficients:

$$Ax+By+C_1=0,Ax+By+C_2=0.$$

• Use

$$d=\frac{|C_1-C_2|}{\sqrt{A^2+B^2}}.$$

12. Constructing the equation of a circle from its centre and radius

• Identify the centre $(h,k)$.
• Identify the radius $r$.
• Substitute into

$$(x-h{)}^2+(y-k{)}^2=r^2.$$

• If the centre is $(0,0)$, use

$$x^2+y^2=r^2.$$

13. Extracting the centre and radius from the general circle equation

Given $$x^2+y^2+2gx+2fy+c=0:$$

• Compare the equation with the general form.
• State the centre as

$$(-g,-f).$$

• State the radius as

$$\sqrt{g^2+f^2-c}.$$

• Check the condition $g^2+f^2-c>0$ for a real circle.
• If $g^2+f^2-c=0$, identify the result as a point circle.

14. Finding where a circle cuts the coordinate axes

For a circle with centre $(h,k)$ and radius $r$:

• To find x-axis intersections, put $y=0$ in

$$(x-h{)}^2+(y-k{)}^2=r^2.$$

• Solve the resulting equation for $x$.
• To find y-axis intersections, put $x=0$.
• Solve the resulting equation for $y$.

15. Finding the circle with a given diameter

Given diameter endpoints $(x_1,y_1)$ and $(x_2,y_2)$:

• Substitute the coordinates into

$$(x-x_1)(x-x_2)+(y-y_1)(y-y_2)=0.$$

• Expand if required.

16. Finding a circle through three non-collinear points

• Start with

$$x^2+y^2+2gx+2fy+c=0.$$

• Substitute each of the three non-collinear points.
• Solve the resulting three equations for $g$, $f$, and $c$.
• Substitute these values back into the general equation.

17. Finding a tangent to a circle

For the origin-centred circle $$x^2+y^2=r^2$$ at $(x_1,y_1)$:

• Confirm that the point lies on the circle.
• Use the tangent equation

$$x{x}_1+y{y}_1=r^2.$$

For a circle with centre $(h,k)$:

• Write the tangent line in the form $Ax+By+C=0$.
• Find the perpendicular distance from $(h,k)$ to the line:

$$\frac{|Ah+Bk+C|}{\sqrt{A^2+B^2}}.$$

• Set this distance equal to $r$.

18. Analysing a line-circle relationship

• Substitute the line equation into the circle equation, or calculate the perpendicular distance from the circle’s centre to the line.
• Interpret the result:

- no real intersection: no intersection; - one intersection: tangent; - two intersections: secant line.

• In the distance method, tangency occurs specifically when the centre-to-line distance equals the radius.

Where It Goes Wrong

• The slope formula is applied to a vertical line without checking whether $x_2-x_1=0$; a vertical line has undefined slope and equation $x=a$.
• The condition for perpendicular lines, $m_1{m}_2=-1$, is used without noting that both slopes must be finite.
• The section formula is used for an internal ratio $m:n$ without preserving the correct weighted order:

$$\left(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n}\right).$$

• The x-intercept and y-intercept of $Ax+By+C=0$ are confused; the x-intercept requires $y=0$, while the y-intercept requires $x=0$.
• The centre of

$$x^2+y^2+2gx+2fy+c=0$$ is incorrectly written as $(g,f)$ rather than $(-g,-f)$, and the radius condition $g^2+f^2-c>0$ is forgotten.

• The tangent equation $x{x}_1+y{y}_1=r^2$ is used without verifying that $(x_1,y_1)$ lies on $x^2+y^2=r^2$; for a general circle, the distance from the centre to the line must equal the radius.

What Gets Asked

This material supports questions requiring students to:

• locate and interpret points in the Cartesian coordinate system;
• calculate distances, midpoints, and internal division points;
• determine the slope or angle of inclination of a line;
• form line equations using slope-intercept, point-slope, two-point, intercept, or general form;
• find x- and y-intercepts from $Ax+By+C=0$;
• test whether lines are parallel or perpendicular;
• calculate the angle between two lines;
• find the distance from a point to a line or between parallel lines;
• construct a circle from its centre and radius;
• identify the centre and radius from $x^2+y^2+2gx+2fy+c=0$;
• determine whether a circle is real or a point circle;
• find where a circle cuts the coordinate axes;
• find the equation of a circle with a specified diameter;
• find the equation of a circle through three non-collinear points;
• find the tangent at a specified point on $x^2+y^2=r^2$;
• test whether a line is tangent to a circle;
• distinguish between no intersection, tangency, and a secant line using substitution or distance.