Mathematics · Class 11 Science
Ch 11Straight Lines — Class 11 Mathematics, concept-first.
Before developing the geometry of the straight line, we briefly recall the coordinate-geometry tools built in earlier classes that this chapter uses directly.
Key concepts
Hover a concept to preview it and jump to its most relevant Q&A.
Angle Between Two Lines
When you think of two lines crossing each other, the first thing you notice is how "wide" or "narrow" the opening between them is. That opening is the angle between the lines.
Most relevant Q&A
- Find the equation of the line passing through the point $(2, 3)$ and perpendicular to the line joining the points $(1, 4)$ and $(5, -2)$.Free
- The line through the points $(h, 3)$ and $(2, 7)$ is perpendicular to the line $5x - 2y - 10 = 0$. Find the value of $h$.Preview
- If the angle between two lines is $45^\circ$ and the slope of one of the lines is $3$, find the slope of the other line. (Two answers are po…Free
- Find the value of $k$ so that the line through $(k, 3)$ and $(2, -4)$ is perpendicular to the line through $(5, 1)$ and $(-1, 7)$.Free
- Show that the line joining $(1, 1)$ and $(3, 5)$ is parallel to the line joining $(2, -1)$ and $(4, 3)$.Preview
Chapter contents
The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.
Brief Recall of 2D Coordinate Geometry
Before developing the geometry of the straight line, we briefly recall the coordinate-geometry tools built in earlier classes that this chapter uses directly.
Slope of a Line
The slope (or gradient) of a line is the single number measuring how steeply it rises or falls, and it is the starting point for everything else in this chapter.
Angle Between Two Lines
When two non-parallel lines intersect, they form two pairs of equal vertically-opposite angles that are supplementary to each other; this section finds those angles directly from the two lines' slopes…
Lines Parallel to the Axes
Before building the general forms of a line's equation, it is worth isolating two special — but very common — cases directly from the definition of a straight line as the set of points satisfying one…
Point-Slope Form
Every remaining form in this chapter is really just a special case, or a short consequence, of the single idea in this section: fixing one point and the slope pins down a line completely, and its equa…
Slope-Intercept Form
This section specializes the point-slope form of Section 5 to the one particular point every non-vertical line is guaranteed to have: the point where it crosses the y-axis.
Two-Point Form
Frequently a line is described not by a point and a slope but by two points it passes through; this section shows the slope-then-point-slope route that handles this case in two short steps.
Intercept Form
This section specializes the two-point form of Section 7 to the case where the two given points are the line's own x- and y-intercepts — the two places it crosses the coordinate axes.
Distance of a Point from a Line
Given a line and a point not on , this section finds the length of the perpendicular dropped from to — the shortest possible distance from the point to the line.
Summary
This chapter developed the algebra of the straight line from its two most basic descriptors — a slope and a point (or two points, or an intercept) — building every named form from the single point-slo…
More questions
27 Q+−Show 3 questionsHide questions3 questions
- Q25Find the equation of the line passing through the point $(2, 3)$ and perpendicular to the line joining the points $(1, 4)$ and $(5, -2)$.Free
- Q26The line through the points $(h, 3)$ and $(2, 7)$ is perpendicular to the line $5x - 2y - 10 = 0$. Find the value of $h$.Preview
- Q27Prove that the line through the points $(2, -3)$ and $(-4, 5)$ is parallel to the line through the points $(5, 1)$ and $(-1, 9)$, and find t…Preview
+−Show 8 questionsHide questions8 questions
- Example 1Find the slope of the line passing through $(4, -3)$ and $(-2, 5)$. Hence determine whether the points $(4, -3)$, $(-2, 5)$ and $(10, -11)$…Free
- Example 2If the angle between two lines is $45^\circ$ and the slope of one of the lines is $3$, find the slope of the other line. (Two answers are po…Free
- Example 3Find the value of $k$ so that the line through $(k, 3)$ and $(2, -4)$ is perpendicular to the line through $(5, 1)$ and $(-1, 7)$.Free
- Example 4Find the equations of the lines through the point $(5, -2)$ that are (a) parallel to the x-axis, and (b) parallel to the y-axis.Preview
- Example 5Find the equation of the line with slope $-3$ passing through the point $(4, -7)$.Preview
- Example 6Find the equation of the line passing through the points $(2, 3)$ and $(-4, -9)$ using the two-point form. Hence write the equation in slope…Preview
- Example 7A line makes intercepts $4$ and $-6$ on the x-axis and y-axis respectively. Find its equation. Verify your answer by finding the x- and y-in…Preview
- Example 8Find the distance of the point $(2, 5)$ from the line $3x - 4y + 8 = 0$. Also find the distance between the parallel lines $3x - 4y + 8 = 0$…Preview
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- Q9Find the slope of the line joining the points $(-3, 2)$ and $(5, -6)$.Free
- Q10Find the angle of inclination of a line whose slope is $\sqrt{3}$.Free
- Q11Show that the line joining $(1, 1)$ and $(3, 5)$ is parallel to the line joining $(2, -1)$ and $(4, 3)$.Preview
- Q12Find the angle between the lines whose slopes are $1$ and $-\dfrac{1}{2}$.Preview
- Q13Using slopes, show that the points $(4, 4)$, $(3, 5)$ and $(-1, -1)$ are the vertices of a right-angled triangle, and state at which vertex…Preview
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- Q14Find the equations of the lines through the point $(-3, 4)$ that are parallel to the x-axis and to the y-axis.Free
- Q15Find the equation of the line with slope $\dfrac{1}{2}$ passing through $(-4, 3)$.Free
- Q16Find the equation of the line passing through $(0, -3)$ and having an inclination of $30^\circ$.Free
- Q17Find the equation of the line passing through the points $(-1, 1)$ and $(2, -4)$.Preview
- Q18Find the equation of the line which makes intercepts $5$ and $3$ on the x-axis and y-axis respectively.Preview
- Q19Reduce the equation $4x - 3y + 12 = 0$ to slope-intercept form, and find its slope and y-intercept.Preview
- Q20Reduce the equation $2x + 3y - 6 = 0$ to intercept form, and find the intercepts made by the line on the axes.Preview
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- Q21Find the distance of the point $(3, -2)$ from the line $5x - 12y + 26 = 0$.Free
- Q22Find the distance between the parallel lines $4x + 3y - 7 = 0$ and $4x + 3y + 8 = 0$.Free
- Q23Find the value(s) of $p$ if the distance of the point $(2, p)$ from the line $3x - 4y - 5 = 0$ is $4$ units.Preview
- Q24Find the distance between the parallel lines $3x + 4y = 9$ and $6x + 8y = 15$. (First check that the two equations really do represent paral…Preview