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Applied Mathematics · Class 11 Commerce

Ch 12Coordinate Geometry — Class 11 Applied Mathematics, concept-first.

This concept map shows how the chapter fits together. It studies three curves through their equations: the straight line (slope-intercept, intercept and normal forms, all reducible to Ax + By + C = 0), the circle (centre-radius, general and diameter forms), and the parabola (the four standard orientations y² = ±4ax and…

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Key concepts

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Straight Line

Before any formula, think about what a straight line does. If you walk from your home to school, the shortest path is a straight line. If you stretch a thread tight between two pins, it forms a straight line.

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Chapter contents

The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.

Concept Map

This concept map shows how the chapter fits together. It studies three curves through their equations: the straight line (slope-intercept, intercept and normal forms, all reducible to Ax + By + C = 0)…

12.1

Straight Line

A straight line is the simplest path between two points, and in coordinate geometry we give it a precise algebraic home.

12.2

Slopes of Perpendicular Lines

Two lines are perpendicular if they meet at a right angle — a relationship that shows up everywhere, from the edges of a square to the axes of a graph.

12.3

Angle Between Two Lines

We often think of lines as having a direction, and the angle between them is simply the measure of how much one line must be rotated to align with the other.

12.4

Various Forms of the Equation of a Line

19 Q

In the previous sections, you learned how to describe a line using its slope and a point, or just its slope and intercept.

+Worked Examplesi9 questions
  1. Example 1Let us assume that the demand curve is described by the line $q = mp + b$. Find its equation given that a promoter discovers that the demand…Free
  2. Example 2The equilibrium quantity and the equilibrium price of a product are determined by the point where the supply and demand curves intersect. Fo…Free
  3. Example 3Find the equations of the lines parallel to the axes and passing through the point $(-3, 5)$.Free
  4. Example 4Find the equation of the perpendicular bisector of the line segment joining the points A$(2, 3)$ and B$(6, -5)$.Preview
  5. Example 5Find the equation of the line whose y-intercept is $-3$ and which is perpendicular to the line $3x - 2y + 5 = 0$.Preview
  6. Example 6If M$(a, b)$ is the midpoint of a line segment intercepted between the axes, show that the equation of the line is $\frac{x}{a} + \frac{y}{b…Preview
  7. Example 7Find the equation of a line whose perpendicular distance from the origin is 2 units and the angle between the perpendicular segment and the…Preview
  8. Example 8Reduce the equation $\sqrt{3}x + y + 2 = 0$ to the normal form $x\cos\alpha + y\sin\alpha = p$ and hence find the value of $\alpha$ and $p$.Preview
  9. Example 9Find the distance between the parallel lines $15x + 8y - 34 = 0$ and $15x + 8y + 31 = 0$.Preview
+Exercise 12.1i10 questions
  1. Q1Find the equation of a line which is equidistant from the lines $y = 8$ and $y = -2$.Free
  2. Q2If A$(1, 4)$, B$(2, -3)$ and C$(-1, -2)$ are the vertices of a $\triangle ABC$, then find the equation of (i) the median through A (ii) the…Free
  3. Q3Find the equation of the bisector of the angle between the coordinate axes.Free
  4. Q4Find the equation of the line passing through the point $(2, 2)$ and cutting off intercepts on the axes, whose sum is 9.Preview
  5. Q5Find the equation of the line which is at a distance of 3 units from the origin such that $\tan\alpha = \frac{5}{12}$, where $\alpha$ is the…Preview
  6. Q6Reduce the equation $5x - 12y = 60$ to intercept form. Hence find the length of the portion of the line intercepted between the axes.Preview
  7. Q7Reduce the equation $x + y - 2 = 0$ to the normal form.Preview
  8. Q8What are the points on the x-axis whose perpendicular distance from the line $\frac{x}{3} + \frac{y}{4} = 1$ is 4 units.Preview
  9. Q9A company produces shoes. When 30 shoes are produced the total cost of production is Rs. 1500. When 50 shoes are produced the costs increase…Preview
  10. Q10Consider a market characterised by the following supply and demand curves: $q_{demand} = -10p + 1000$ $q_{supply} = 0.2p + 2986$ Find the eq…Preview
12.5

Circle

A circle is simply the set of all points that lie at a fixed distance from a given centre. That fixed distance is the radius, and the centre is the anchor that defines the shape.

12.6

Properties of Circles

12 Q

A circle is defined by a fixed centre and a constant radius, but its true character emerges when you examine the relationships between its points, chords, and tangents.

+Worked Examplesi5 questions
  1. Example 10Find the equation of a circle with centre $(3, -2)$ and radius 5.Free
  2. Example 11Find the equation of the circle with centre $(2, 2)$ and which passes through the point $(4, 5)$.Free
  3. Example 12Find the equation of the circle of radius 5 whose centre lies on x-axis and passes through the point $(2, 3)$.Preview
  4. Example 13Find the equation of the circle passing through $(0, 0)$ and which makes intercepts $a$ and $b$ on the coordinate axes.Preview
  5. Example 14Find the centre and radius of the circle $x^2 + y^2 - 8x + 10y - 12 = 0$.Preview
+Exercise 12.2i7 questions
  1. Q1Find the equation of the circle with: (i) centre $(0, 2)$ and radius 2. (ii) centre $(0, 0)$ and radius 3. (iii) centre $(-a, -b)$ and radiu…Free
  2. Q2Find the equation of the circle drawn on a diagonal of the rectangle as its diameter whose sides are the lines $x = 4$, $x = -5$, $y = 5$ an…Free
  3. Q3Which of the following equations represent a circle? If so, determine its centre and radius. (i) $3x^2 + 3y^2 + 6x - 4y = 1$ (ii) $2x^2 + 2y…Free
  4. Q4One end of a diameter of the circle $x^2 + y^2 - 6x + 5y - 7 = 0$ is $(-1, 3)$. Find the coordinates of the other end of the diameter.Preview
  5. Q5Find the equation of the circle passing through the points $(2, 3)$ and $(-1, 1)$ and whose centre lies on the line $x - 3y - 11 = 0$.Preview
  6. Q6Find the value of $p$ so that $x^2 + y^2 + 8x + 10y + p = 0$ is the equation of a circle of radius 7 units.Preview
  7. Q7Find the value of $k$ for which the circles $x^2 + y^2 - 3x + ky - 5 = 0$ and $4x^2 + 4y^2 - 12x - y - 9 = 0$ are concentric.Preview
12.7

Parabola

A parabola is the set of all points that are equidistant from a fixed point (the focus) and a fixed straight line (the directrix).

12.8

Length of the Latus Rectum

9 Q

The latus rectum is a special chord through a focus, perpendicular to the axis. Its length is a fixed number for a given parabola, independent of which focus you choose.

+Worked Examplesi4 questions
  1. Example 15If a parabolic reflector is 20 cm in diameter and 5 cm deep, find the focus.Free
  2. Example 16A beam is supported at its ends by supports which are 12 metres apart. Since the load is concentrated at the centre of the beam there is a d…Free
  3. Example 17The girder of a railway bridge is a parabola with its vertex at the highest point 10 metre above the ends. If the span is 100 metres, find t…Preview
  4. Example 18The cable of a uniformly loaded bridge hangs in the form of a parabola. The roadway which is horizontal and 100 m long is supported by verti…Preview
+Exercise 12.3i5 questions
  1. Q1The focus of the parabolic mirror is at a distance of 5 cm from its vertex. If the mirror is 45 cm deep, find the distance AB (the width of…Free
  2. Q2An arc is in the form of a parabola with its axis vertical. The arc is 10 m high and 5 m wide at the base. How wide is it 2 m from the verte…Free
  3. Q3The towers of a suspension bridge, hang in the form of a parabola, have their tops 30 metres above the roadway and are 200 metres apart. If…Preview
  4. Q4The girder of a railway bridge is in the form of a parabola with its vertex at the highest point, 15 metres above the ends. If the span is 1…Preview
  5. Q5A water jet from a fountain reaches its maximum height of 4 metres at a distance of 0.5 metres from the vertical passing through the point O…Preview