Mathematics and Statistics · Class 11 Commerce
Ch 5Locus and Straight Line — Class 11 Mathematics and Statistics, concept-first.
In everyday language a path is the trail traced out by a moving object — the arc of a thrown ball, the circle swept by the tip of a clock hand. In coordinate geometry we make this idea precise. A locus (plural loci) is the set of all points in a plane that satisfy one stated geometric condition, and only those points.
Key concepts
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Locus and Its Equation
A locus is the algebraic translation of a geometric rule that a moving point obeys. The working method is always the same: name the moving point , state its distance condition(s) using the distance formula , square to cl…
Most relevant Q&A
- Find the equation of the locus of a point which is equidistant from the points $A(1, 2)$ and $B(3, 4)$.Free
- A point $P$ moves in the plane so that it is always equidistant from the two fixed points $A(2, 3)$ and $B(4, 1)$. Find the equation of the…Free
- A point $P$ moves so that its distance from the fixed point $C(1, 2)$ is always $5$ units. Find the equation of its locus.Free
Chapter contents
The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.
Locus and Its Equation
In everyday language a path is the trail traced out by a moving object — the arc of a thrown ball, the circle swept by the tip of a clock hand. In coordinate geometry we make this idea precise.
Slope (Gradient) of a Line
The slope (or gradient) of a straight line measures its steepness and direction of rise. It is denoted by and defined through the angle the line makes with the positive direction of the x-axis.
Standard Forms of the Equation of a Straight Line
A straight line is fixed by knowing enough about it — a point and a direction, or two points, or its intercepts. Each kind of information leads to a convenient standard form.
The General Form of a Line
Every straight line in the plane — including vertical lines that have no slope — can be written in one common form:
Parallel and Perpendicular Lines
The slope captures a line's direction, so questions about two lines being parallel or perpendicular reduce to simple statements about their slopes.
Angle Between Two Lines
When two lines cross, they form two pairs of equal angles (one acute, one obtuse, adding to ). If the lines have slopes and , the angle between them is given by the tangent formula:
Distance of a Point from a Line
The distance of a point from a line means the shortest distance — the length of the perpendicular dropped from the point onto the line.
Exercises
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- Q15Find the equation of the locus of a point which is equidistant from the points $A(1, 2)$ and $B(3, 4)$.Free
- Q16Find the slope of the line passing through the points $(-1, 2)$ and $(3, -6)$.Free
- Q17Find the equation of the line having x-intercept $2$ and y-intercept $-5$.Free
- Q18Find the equation of the line passing through $(4, 5)$ and parallel to the line $2x - y + 3 = 0$.Preview
- Q19Find the perpendicular distance of the point $(2, 3)$ from the line $4x + 3y - 10 = 0$.Preview
- Q20Reduce the equation $2x + 3y - 6 = 0$ to (i) slope-intercept form and (ii) intercept form, and hence state its slope and both intercepts.Preview
- Q21Find the acute angle between the line $y = x$ and the x-axis.Preview
More questions
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- Example 1A point $P$ moves in the plane so that it is always equidistant from the two fixed points $A(2, 3)$ and $B(4, 1)$. Find the equation of the…Free
- Example 2A point $P$ moves so that its distance from the fixed point $C(1, 2)$ is always $5$ units. Find the equation of its locus.Free
- Example 3Find the slope of the line passing through the points $A(2, 3)$ and $B(5, 9)$.Free
- Example 4Find the slope of a line whose inclination with the positive x-axis is $135^\circ$.Preview
- Example 5Find the equation of the line which passes through the point $(1, 4)$ and has slope $2$.Preview
- Example 6Find the equation of the line passing through the points $(1, 2)$ and $(3, 8)$.Preview
- Example 7Find the equation of the line which makes an intercept of $4$ on the x-axis and an intercept of $3$ on the y-axis.Preview
- Example 8Find the equation of the line, the length of whose perpendicular from the origin is $5$ units, and the perpendicular makes an angle of $30^\…Preview
- Example 9Find the equation of the line passing through the point $(2, 3)$ and parallel to the line $4x + 5y = 6$.Preview
- Example 10Find the equation of the line passing through $(1, -2)$ and perpendicular to the line $3x - 4y + 5 = 0$.Preview
- Example 11Find the acute angle between the two lines whose slopes are $3$ and $-2$.Preview
- Example 12Show that the lines $3x + 2y = 5$ and $2x - 3y = 7$ are perpendicular to each other.Preview
- Example 13Find the perpendicular distance of the point $P(3, -5)$ from the line $3x - 4y - 26 = 0$.Preview
- Example 14Find the distance between the two parallel lines $3x + 4y + 7 = 0$ and $3x + 4y - 8 = 0$.Preview