## Coordinate Geometry Maths Notes

**Read and Understand**

**In the previous standard, we understood, how to find the distance between any two points on a number line.**

Here coordinate of point P is 3 and coordinate of point Q is 5.

∴ d(P,Q) = 5 – 3 = 2

If coordinate of point Q is x_{2} and coordinate of point P is x_{1} and x_{2} > x_{1}, then d(P, Q) = x_{2} – x_{1}.

Using the same concept, we will understand to find the distance between two points in the XY-plane.

(i) To find the distance between any two points on any axis:

In the figure, points P(x_{1}, 0) and Q(x_{2}, 0) lie on X-axis. Q lies to the right side of P.

∴ X_{2} > X_{1}

∴ d(P, Q) = x_{2} – x_{1}.

(ii) To find distance between any two po1nts on Y-axis.

In the figure, points R(0, y_{1}) and S(0, y_{2}) lie on Y-axis.

The points S lie above point R.

∴ y_{2} > y_{1}

∴ d(R, S) = y_{2} – y_{1}

**To find the distance between the two points if the segment joining these points is parallel to any axis in the XY-plane:**

(i) In the figure, seg KT is parallel to X-axis.

∴ y-coordinates of points K and T are equal.

We draw seg KP and seg TQ perpendicular to X-axis.

∴ ▢ KPQT is a rectangle.

∴ KT = PQ … (Opposite sides of a rectangle)

as PQ = x_{2} – x_{1},

KT = x_{2} – x_{1}

(ii)

In the figure, seg CD is parallel to Y-axis.

∴ x-coordinates of points C and D are equal.

Draw seg CS and segDR perpeidicuIar to Y-axis.

∴ ▢CSRD is a rectangle.

∴ CD = SR … (Opposite sides of a rectangle)

as SR = y_{2} – y_{1}, CD = y_{2} – y_{1}

**Distance formula:**

In the figure, P(x_{1}, y_{1}) and Q(x_{2}, y_{2}) are any two points in the XY-plane.

From point Q draw perpendicular QS on X-axis.

Similarly from point P draw perpendicular PT on seg QS.

seg QS is parallel to Y-axis.

∴ the x-coordinate of point T is x_{2}.

seg PT is parallel to X-axis.

∴ the y-coordinate of point T is y_{1}.

∴ PT = d(P,T) = x_{2} – x_{1};

QT = d(Q, T) = y_{2} – y_{1}

In right angled △ PTQ.

PQ^{2} + QT^{2} … (Pythagoras theorem)

= (x_{2} – x_{1})^{2} + (y_{2} – y_{1})^{2}

∴ PQ = \(\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}\)

This is known as distance formula.

Note that,

\(\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}\) = \(\sqrt{\left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}}\)

**Remember this!**

- Coordinates of origin are (0, 0). Hence if coordinates of point A are (x, y), then d(O, A) = \(\sqrt{x^{2}+y^{2}}\)
- If points A(x
_{1}, y_{1}), B(x_{2}, y_{2}) 1i in the XY-plane, then d(A, B) = \(\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}\)

that is,

AB^{2}= (x_{2}– x_{1})^{2}+ (y_{2}– y_{1})^{2}= (x_{1}– x_{2})^{2}+ (y_{1}– y_{2})^{2}.

**Division of a line segment:**

In the figure, XZ = 4 and ZY = 6

∴ \(\frac{\mathrm{XZ}}{\mathrm{ZY}}=\frac{4}{6}=\frac{2}{3}\)

∴ we can say that, ‘Point Z divides the line segment XY in the ratio 2 : 3’.

**Section Formula:**

In the figure, point P on the seg RS in XY-plane, divides seg RS in the ratio m : n.

Assume R(x_{1}, y_{1}), S(x_{2}, y_{2}) and P(x, y). Draw seg RT, seg PQ and seg SM perpendicular to X-axis.

∴ T(x_{1}, 0); Q(x, 0) and M(x_{2}, 0).

∴ TQ = x – x_{1} and QM = x_{2} – x ……… (1)

seg RT || seg PQ seg SM.

∴ by the property of intercepts made by three parallel lines, \(\frac{\mathrm{RP}}{\mathrm{PS}}=\frac{\mathrm{TQ}}{\mathrm{QM}}=\frac{m}{n}\)

Now TQ = x – x_{1} and QM = x_{2} – x … [From (1)1

∴ \(\frac{x-x_{1}}{x_{2}-x}=\frac{m}{n}\)

∴ n(x – x_{1}) = m(x_{2} – x)

∴ nx – nx_{1} = mx_{2} – mx

∴ mx + nx = mx_{2} + nx_{1}

∴ x(m + n) = mx_{2} + nx_{1}

∴ x = \(\frac{m x_{2}+n x_{1}}{m+n}\)

Similarly drawing perpendiculars from points R, P and S to Y-axis, we get, y = \(\frac{m y_{2}+n y_{1}}{m+n}\)

∴ coordinates of the point, which divides the line

segment joining the points R(x_{1}, y_{1}) and S(x_{2}, y_{2}) in the ratio m : n are given by

\(\left(\frac{m x_{2}+n x_{1}}{m+n}, \frac{m y_{2}+n y_{1}}{m+n}\right)\)

**Coordinates of the midpoint of a segment:**

If P(x_{1}, y_{1}) and Q(x_{2}, y_{2}) are two points and M(x, y) is the midpoint of seg PQ then m = n.

**Let’s recall**

- The medians of a triangle are concurrent.
- The point of concurrence of the medians is called centroid.
- Centroid divides the median in the ratio 2: 1.

**Centroid Formula:**

Suppose, in ∆PQR, P(x_{1}, y_{1}), Q(x_{2}, y_{2}) and R(x_{3}, y_{3}) are the vertices. Seg PM is a median and G(x, y) is the centroid.

By definition of the median, M is the midpoint of seg QR.

Coordinates of M, by midpoint formula is

\(\left(\frac{x_{2}+x_{3}}{2}, \frac{y_{2}+y_{3}}{2}\right)\)

G(x, y) is the centroid of ∆ PQR

∴ PG : GM = 2 : 1

By section formula,

Thus, if (x_{1}, y_{1}), (x_{2}, y_{2}) and (x_{3}, y_{3}) are the vertices of a triangle then the coordinates of the centroid are

\(\left(\frac{x_{1}+x_{2}+x_{3}}{3}, \frac{y_{1}+y_{2}+y_{3}}{3}\right)\).

This is called centroid formula.

**Remember this!**

**Section formula:**

The coordinates of a point which divides the line segment joined by two distinct points (x_{1}, y_{1}) and (x_{2}, y_{2}) in the ratio m : n are

\(\left(\frac{m x_{2}+n x_{1}}{m+n}, \frac{m y_{2}+n y_{1}}{m+n}\right)\)

**Midpoint formula:**

The coordinates of midpoint of a line segment

joining two distinct points (x_{1}, y_{1}) and (x_{2}, y_{2}) are

\(\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\right)\)

**Centroid formula:**

If (x_{1}, y_{1}), (x_{2}, y_{2}) and (x_{3}, y_{3}) are the vertices of a triangle then coordinates of the centroid are

\(\left(\frac{x_{1}+x_{2}+x_{3}}{3}, \frac{y_{1}+y_{2}+y_{3}}{3}\right)\)

**Slope of a Line:**

In the figure P(x_{1}, y_{1}) and Q(x_{2}, y_{2}) are any two paints in XY-plane.

the ratio \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\) is constant.

It is true for any two points on line l.

The ratio \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\) is called the slope of the line l.

Usually slope is denoted by letter m.

∴ m = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)

**Remember this!**

- Slope of X-axis and any line parallel to X-axis is O.
- Slope of Y-axis and any line parallel to Y-axis cannot be determined.
- Parallel lines have equal slopes.
- When two distinct fines have sanie slope, then the two lines are parallel.

**Slope of line-using ratio in trigonometry**

The tangent ratio of an angle made by the line with the positive direction of X-axis is called the slope of that line.

In the figure, line l makes an angle θ with the positive direction of X-axis.

∴ slope of line l = m = tan θ.