Coordinate Geometry formulas for JEE Main & Advanced

23 formula entries for Coordinate Geometry, in the order the chapter builds them. Each shows the formula, when to use it and the first trap to avoid.

Coordinate System & Distance Formula

P Q = \sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}}

When to use. Use to find the length of a line segment between two points, or to check collinearity (sum of two distances equals third).

Trap. Forgetting to square the differences before adding.

Section Formula

Internal:

P = (\frac{m x_{2} + n x_{1}}{m + n}, \frac{m y_{2} + n y_{1}}{m + n})

; External:

P = (\frac{m x_{2} - n x_{1}}{m - n}, \frac{m y_{2} - n y_{1}}{m - n})

When to use. Use to find coordinates of a point dividing a segment in a given ratio, or to find centroid, incenter, or harmonic conjugates.

Trap. Mixing up $m$ and $n$ with the points; for $A P : P B = m : n$ , $m$ multiplies $B$ and $n$ multiplies $A$ (each point is weighted by the opposite segment).

Area of a Triangle

Δ = \frac{1}{2} ∣ x_{1} (y_{2} - y_{3}) + x_{2} (y_{3} - y_{1}) + x_{3} (y_{1} - y_{2}) ∣

When to use. Use to compute area from coordinates, or to check collinearity (area = 0).

Trap. Forgetting the absolute value; area is always positive.

Slope of a Line

m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \tan θ

When to use. Use to find line equations, check parallelism/perpendicularity, or find angle between lines.

Trap. Dividing by zero when $x_{1} = x_{2}$ (vertical line, slope undefined).

Equation of a Line: Various Forms

y = m x + c

;

y - y_{1} = m (x - x_{1})

;

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

;

A x + B y + C = 0

When to use. Use appropriate form based on given data: slope and intercept, point and slope, two points, or intercepts.

Trap. Using slope-intercept for vertical lines (slope undefined).

Angle Between Two Lines

\tan θ = ∣ \frac{m_{1} - m_{2}}{1 + m_{1} m_{2}} ∣

When to use. Use to find acute angle between two lines, or to check if lines are parallel or perpendicular.

Trap. Forgetting the absolute value; angle is acute unless specified.

Distance from a Point to a Line

d = \frac{∣ A x_{1} + B y_{1} + C ∣}{\sqrt{A^{2} + B^{2}}}

When to use. Use to find shortest distance from a point to a line, or in locus problems.

Trap. Forgetting the absolute value in numerator.

Distance Between Two Parallel Lines

d = \frac{∣ C_{1} - C_{2} ∣}{\sqrt{A^{2} + B^{2}}}

When to use. Use to find distance between two parallel lines, e.g., in problems involving strips or loci.

Trap. Forgetting that lines must be in the same form (coefficients of $x$ and $y$ proportional).

Locus

No single formula; derived from condition.

When to use. Use to find equation of a curve defined by a geometric property (e.g., equidistant from two points).

Trap. Forgetting to eliminate parameters when multiple conditions exist.

Circle: Standard Equation

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

When to use. Use to represent a circle given center and radius, or to find center and radius from general form.

Trap. Forgetting to square the radius.

Equation of Circle: Diameter Form

(x - x_{1}) (x - x_{2}) + (y - y_{1}) (y - y_{2}) = 0

When to use. Use when given endpoints of a diameter to quickly write circle equation.

Trap. Forgetting that this is derived from the condition that angle in a semicircle is 90°.

Tangent to a Circle

T = 0

x x_{1} + y y_{1} + g (x + x_{1}) + f (y + y_{1}) + c = 0

When to use. Use to find equation of tangent at a given point on the circle.

Trap. Forgetting to replace $x^{2}$ with $x x_{1}$ , $y^{2}$ with $y y_{1}$ , $x$ with $\frac{x + x_{1}}{2}$ , etc.

Length of Tangent from External Point

L = \sqrt{S_{1}}

where

S_{1} = x_{1}^{2} + y_{1}^{2} + 2 g x_{1} + 2 f y_{1} + c

When to use. Use to find length of tangent from an external point to a circle.

Trap. Forgetting that $S_{1}$ must be positive for external point.

Parabola: Standard Equation

y^{2} = 4 a x

(right);

y^{2} = - 4 a x

(left);

x^{2} = 4 a y

(up);

x^{2} = - 4 a y

(down)

When to use. Use to identify parabola orientation and key parameters from equation.

Trap. Confusing focus and directrix: for $y^{2} = 4 a x$ , focus is $(a, 0)$ , not $(0, a)$ .

Tangent to Parabola $y^{2} = 4 a x$

y y_{1} = 2 a (x + x_{1})

;

y = m x + \frac{a}{m}

When to use. Use to find equation of tangent to parabola at a point or with given slope.

Trap. Forgetting the factor of 2 in $y y_{1} = 2 a (x + x_{1})$ .

Ellipse: Standard Equation

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1

;

e = \sqrt{1 - \frac{b^{2}}{a^{2}}}

When to use. Use to represent ellipse and find foci, vertices, eccentricity.

Trap. Confusing $a$ and $b$ : $a$ is semi-major axis (larger denominator under $x^{2}$ if horizontal).

Tangent to Ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$

\frac{x x_{1}}{a^{2}} + \frac{y y_{1}}{b^{2}} = 1

;

y = m x \pm \sqrt{a^{2} m^{2} + b^{2}}

When to use. Use to find tangent equation to ellipse at a point or with given slope.

Trap. Forgetting the plus/minus in slope form; two tangents for a given slope.

Hyperbola: Standard Equation

\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1

;

e = \sqrt{1 + \frac{b^{2}}{a^{2}}}

; asymptotes:

y = \pm \frac{b}{a} x

When to use. Use to represent hyperbola and find foci, vertices, asymptotes.

Trap. Confusing with ellipse: hyperbola has minus sign and $e > 1$ .

Tangent to Hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$

\frac{x x_{1}}{a^{2}} - \frac{y y_{1}}{b^{2}} = 1

;

y = m x \pm \sqrt{a^{2} m^{2} - b^{2}}

When to use. Use to find tangent equation to hyperbola at a point or with given slope.

Trap. Forgetting the minus sign in point form.

Rectangular Hyperbola

x y = c^{2}

(rotated form);

x^{2} - y^{2} = a^{2}

When to use. Use when hyperbola has equal axes or equation $x y = c^{2}$ .

Trap. Confusing $x y = c^{2}$ with standard hyperbola; it's rotated 45°.

Condition for Tangency: Line to Conic

For parabola

y^{2} = 4 a x

c = a / m

; for ellipse:

c^{2} = a^{2} m^{2} + b^{2}

; for hyperbola:

c^{2} = a^{2} m^{2} - b^{2}

When to use. Use to find condition for a line to be tangent to a conic, or to find tangent lines with given slope.

Trap. Forgetting to check that the line is not parallel to an asymptote for hyperbola.

Pole and Polar

For circle:

x x_{1} + y y_{1} + g (x + x_{1}) + f (y + y_{1}) + c = 0

When to use. Use to find polar of a point with respect to a conic, or to find pole of a line.

Trap. Forgetting that the polar is a line, not a point.

Chord of Contact