① Define coordinate plane (Cartesian plane).
The plane formed by two straight lines perpendicular to each other is called coordinate plane and the lines \(XOX'\) and \(YOY'\) are called coordinate axes.
Unit 7: Coordinate Geometry
Class 9 Mathematics | Punjab Curriculum and Textbook Board Syllabus 2025
② Define ordered pair.
An ordered pair of real numbers \(x\) and \(y\) is a pair \((x,\ y)\) in which elements are written in specific order.
Note: \((x,\ y) \neq (y,\ x)\)
③ Define origin.
The point of intersection of two coordinate axes is called origin.
④ Differentiate between abscissa and ordinate.
The \(x\)-coordinate of the point is called abscissa of the point \(P(x,y)\) and the \(y\)-coordinate is called its ordinate.
⑤ What are the coordinates of points lying on the axes?
- On the x-axis: \((a, 0)\)
- On the y-axis: \((0, b)\)
⑥ Define distance formula.
If \(A(x_1, y_1)\) and \(B(x_2, y_2)\) are two points and \(d\) is the distance between them, then
\[d = |\overline{AB}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]
\(d \geq 0\)
Note: \(|\overline{AB}|\) stands for \(m\overline{AB}\)
⑦ Define midpoint formula.
If \(A(x_1, y_1)\) and \(B(x_2, y_2)\) are two points in the plane, then the mid-point \(M(x,y)\) of line segment \(\overline{AB}\) is
\[M(x,y) = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\]
⑧ What is the inclination of a line?
The inclination of a line is the angle \(\alpha\) (\(0^\circ < \alpha < 180^\circ\)) measured counterclockwise from the positive \(x\)-axis to a non-horizontal straight line \(\ell\).
- If the line \(\ell\) is parallel to the x-axis, then \(\alpha = 0^\circ\)
- If the line \(\ell\) is parallel to the y-axis, then \(\alpha = 90^\circ\)
⑨ What is the slope or gradient of a line?
Slope or gradient of an inclined path is a measure of its steepness, denoted by \(m\). It is defined as the ratio of rise to run:
\[m = \frac{\text{rise}}{\text{run}} = \frac{y}{x} = \tan\alpha\]
In analytical geometry, for a non-vertical line with inclination \(\alpha\),
\[m = \tan\alpha\]
⑩ What are the special cases of slope?
- If \(\ell\) is horizontal, its slope is zero.
- If \(\ell\) is vertical, its slope is undefined.
- If \(0^\circ < \alpha < 90^\circ\), then \(m\) is positive.
- If \(90^\circ < \alpha < 180^\circ\), then \(m\) is negative.
⑪ What is the formula for the slope of a line passing through two points?
Theorem: If a non-vertical line \(\ell\) with inclination \(\alpha\) passes through two points \(P(x_1, y_1)\) and \(Q(x_2, y_2)\), then the slope or gradient \(m\) of the line is given by:
\[m = \frac{y_2 - y_1}{x_2 - x_1} = \tan\alpha\]
Theorem 2: Let two lines \(\ell_1\) and \(\ell_2\) have slopes \(m_1\) and \(m_2\), respectively:
- Lines are parallel if and only if \(m_1 = m_2\)
- Lines are perpendicular if and only if \(m_1 = -\frac{1}{m_2}\) or \(m_1 \cdot m_2 = -1\)
⑫ Show that the points \(A(-3, 6)\), \(B(3, 2)\), and \(C(6, 0)\) are collinear.
\[\text{Slope of } \overline{AB} = \frac{y_2 - y_1}{x_2 - x_1}\]
\[= \frac{2 - 6}{3 - (-3)}\]
\[= \frac{-4}{6} = \frac{-2}{3}\]
\[\text{Slope of } \overline{BC} = \frac{y_2 - y_1}{x_2 - x_1}\]
\[= \frac{0 - 2}{6 - 3}\]
\[= \frac{-2}{3}\]
\[\because \text{Slope of } \overline{AB} = \text{Slope of } \overline{BC}\]
Hence, the points \(A, B,\) and \(C\) are collinear.
⑬ Show that the triangle with vertices \(A(1, 1)\), \(B(4, 5)\), and \(C(12, -1)\) is a right triangle.
\[\text{Slope of } \overline{AB} = m_1 = \frac{y_2 - y_1}{x_2 - x_1}\]
\[= \frac{5 - 1}{4 - 1}\]
\[m_1 = \frac{4}{3}\]
\[\text{Slope of } \overline{BC} = m_2 = \frac{y_2 - y_1}{x_2 - x_1}\]
\[= \frac{-1 - 5}{12 - 4}\]
\[= \frac{-6}{8}\]
\[m_2 = \frac{-3}{4}\]
\[m_1 \cdot m_2 = \left( \frac{4}{3} \right) \left( \frac{-3}{4} \right)\]
\[m_1 \cdot m_2 = -1\]
Therefore, \(\overline{AB} \perp \overline{BC}\). So \(\triangle ABC\) is a right triangle.
⑭ What is the equation of a straight line parallel to the x-axis (or perpendicular to the y-axis)?
A line that goes left to right (horizontal) is called a line parallel to the x-axis. It is also said to be perpendicular to the y-axis. The equation of this line is:
\[y = a\]
⑮ What is the equation of a straight line parallel to the y-axis (or perpendicular to the x-axis)?
A line that goes up and down (vertical) is called a line parallel to the y-axis. It is also said to be perpendicular to the x-axis. The equation of this line is:
\[x = b\]
⑯ What are the intercepts of a straight line?
The x-intercept of a line is the point where it crosses the x-axis. If the line crosses the x-axis at \((a,0)\), then \(x\)-intercept \(= a\).
The y-intercept of a line is the point where it crosses the y-axis. If the line crosses the y-axis at \((0,b)\), then \(y\)-intercept \(= b\).
⑰ What is the slope-intercept form of a straight line?
The equation of a non-vertical straight line with slope \(m\) and \(y\)-intercept \(c\) is:
\[y = mx + c\]
Note: If the line passes through the origin, then \(c = 0\). So, the equation becomes: \(y = mx\).
⑱ What is the Point-Slope Form of the Equation of a Straight Line?
If a non-vertical line with slope \(m\) passes through a point \(Q(x_1, y_1)\), then its equation is:
\[y - y_1 = m(x - x_1)\]
⑲ What is the equation of a non-vertical straight line passing through two points \(Q(x_1, y_1)\) and \(R(x_2, y_2)\)?
The equation of the line passing through two given points can be derived using the point-slope form:
Step 1: Calculate the slope (m) between the two points:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
Step 2: Using point-slope form with point Q:
\[y - y_1 = m(x - x_1)\]
\[y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1)\]
Alternative Form: Using point R instead:
\[y - y_2 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_2)\]
Key Notes:
- This formula only works for non-vertical lines (where \(x_1 \neq x_2\))
- Both forms are equivalent - they will simplify to the same equation
- You can use either point (Q or R) as the reference point
Example: Find the equation through points (2,3) and (4,7)
\[m = \frac{7-3}{4-2} = 2\]
Using first point: \[y - 3 = 2(x - 2)\]
Simplifies to: \[y = 2x - 1\]
⑳ What is the equation of a non-vertical straight line passing through two points \(Q(x_1, y_1)\) and \(R(x_2, y_2)\)?
The equation of the line passing through two given points is:
\[y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1)\]
or
\[y - y_2 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_2)\]
㉑ What is the equation of a line whose non-zero x-intercept is \(a\) and y-intercept is \(b\)?
The equation of a line with non-zero \(x\)-intercept \(a\) and \(y\)-intercept \(b\) is:
\[\frac{x}{a} + \frac{y}{b} = 1\]
㉒ What is the equation of a non-vertical straight line \(\ell\), where the length of the perpendicular from the origin to \(\ell\) is \(p\), and \(\alpha\) is the inclination of this perpendicular?
The equation of the line is:
\[x\cos\alpha + y\sin\alpha = p\]
㉓ How can the general equation of a line \(ax + by + c = 0\) be transformed into standard forms?
i. Slope-Intercept Form
Starting with general form:
\[ax + by + c = 0\]
Solve for y:
\[by = -ax - c\]
\[y = \frac{-ax - c}{b}\]
\[y = -\frac{a}{b}x - \frac{c}{b}\]
Compare with \(y = mx + c_1\):
\[m = -\frac{a}{b}\]
\[c_1 = -\frac{c}{b}\]
ii. Point-Slope Form
From slope-intercept form, we have slope:
\[m = -\frac{a}{b}\]
Take a known point on the line (x-intercept):
\[Q\left(-\frac{c}{a}, 0\right)\]
Using point-slope form:
\[y - y_1 = m(x - x_1)\]
\[y - 0 = -\frac{a}{b}\left(x - \left(-\frac{c}{a}\right)\right)\]
\[y = -\frac{a}{b}\left(x + \frac{c}{a}\right)\]
\[y = -\frac{a}{b}x - \frac{c}{b}\]
iii. Intercept Form
Starting with general form:
\[ax + by + c = 0\]
Move constant term:
\[ax + by = -c\]
Divide by -c:
\[\frac{ax}{-c} + \frac{by}{-c} = 1\]
Simplify:
\[\frac{x}{-\frac{c}{a}} + \frac{y}{-\frac{c}{b}} = 1\]
Where:
x-intercept \(= -\frac{c}{a}\)
y-intercept \(= -\frac{c}{b}\)
iv. Normal Form
Starting with general form:
\[ax + by + c = 0\]
Divide by \(\pm\sqrt{a^2 + b^2}\):
\[\frac{ax}{\pm\sqrt{a^2 + b^2}} + \frac{by}{\pm\sqrt{a^2 + b^2}} = \frac{-c}{\pm\sqrt{a^2 + b^2}}\]
Let:
\[\cos\alpha = \frac{a}{\pm\sqrt{a^2 + b^2}}\]
\[\sin\alpha = \frac{b}{\pm\sqrt{a^2 + b^2}}\]
\[p = \frac{-c}{\pm\sqrt{a^2 + b^2}}\]
Resulting in normal form:
\[x\cos\alpha + y\sin\alpha = p\]
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