Coordinate Geometry

Course- Diploma Semester-II

Subject- Advanced Mathematics Unit- I

RAI UNIVERSITY, AHMEDABAD

Unit-I COORDINATE GEOMETRY

RAI UNIVERSITY, AHMEDABAD

Introduction— The analytical geometry is also known as coordinate geometry, or Cartesian geometry. It is the study of geometry using a coordinate system. Application— Analytical geometry is widely used in physics and engineering, and is the foundation of most modern fields of geometry, including algebraic, differential, discrete, and computational geometry. Usually the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and squares, often in two and sometimes in three dimensions. Geometrically, one studies the Euclidean plane (2 dimensions) and Euclidean space (3 dimensions). Cartesian coordinate system— Cartesian coordinate system is of two types—

1. Euclidean plane (2 dimensions) or 2-dimensional coordinate system or Plane geometry 2. Euclidean space (3 dimensions) or 3-dimensional coordinate system or Solid geometry

Plane geometry—A two dimensional coordinate system is as shows in the figure given below—There are two perpendicular lines. The horizontal line is called as 풙 − 퐚퐱퐢퐬 and the vertical line is called as 풚 − 퐚퐱퐢퐬. Each of these axis is devide in unity of it. The point of intersection of coordinate axis is a point called as origin. 1.1.1 Point— A point in coordinate geometry is represented by an ordered pair 푃(푥, 푦) of numbers. It is an exact location in the coordinate plane and denoted by a dot. Where 푥 and 푦 represents the perpendicular distance of point from 푦 − axis and 푥 − axis respectively as shown— 1.1.2 Line segment, Line and Ray— If we join two points in such a way that distance between that points become minimum, the that geometrical structure is known as Line segment. If we extend this line segment indefinitely from both ends, it is called as Line. If we extend the line segment indefinitely from one end keeping another end fixed, then such geometrical structure is called as Ray.

2-dimensional coordinate system

Representation of a point on coordinate system

Line Segment, Ray and Line

Unit-I COORDINATE GEOMETRY

RAI UNIVERSITY, AHMEDABAD

1.1.3 Distance between two points— Distance between two points A(푥 ,푦 ) and B(푥 ,푦 ) is given by—

풅 = (풙ퟐ − 풙ퟏ)ퟐ + (풚ퟐ − 풚ퟏ)ퟐ 1.1.4 Dividing a line segment in the ratio m: n— Let a Point 퐑(풙,풚) divides the line segment PQ in ratio m: n, Where, Points P and Q are P ≡ (푥 ,푦 ) and Q ≡ (푥 ,푦 ) Internal Division—

풙 =풎풙ퟐ + 풏풙ퟏ풎 + 풏

풚 =풎풚ퟐ + 풏풚ퟏ풎 + 풏

External Division—

풙 =풎풙ퟐ − 풏풙ퟏ풎− 풏

풚 =

풎풚ퟐ − 풏풚ퟏ풎−풏

Mid Point—

풙 =풙ퟏ + 풙ퟐ

ퟐ

풚 =풚ퟏ + 풚ퟐ

ퟐ

Distance between two points

Internal Division

External Division

Mid Point

External Division

Unit-I COORDINATE GEOMETRY

RAI UNIVERSITY, AHMEDABAD

1.1.5 Area of the triangle—The area of the triangle having vertices A(푥 ,푦 ), B(푥 ,푦 ) and C(푥 ,푦 ) is given by—

1.1.6 Centroid— Let 퐀(풙ퟏ,풚ퟏ), 퐁(풙ퟐ,풚ퟐ) and 퐂(풙ퟑ,풚ퟑ) are vertices of a triangle, Then its Centroid is the point of intersection of the Medians. It is given by—

풙ퟎ =풙ퟏ + 풙ퟐ + 풙ퟑ

ퟑ

풚ퟎ =풚ퟏ + 풚ퟐ + 풚ퟑ

ퟑ 1.1.7 Incenter— Let 퐀(풙ퟏ,풚ퟏ), 퐁(풙ퟐ,풚ퟐ) and 퐂(풙ퟑ,풚ퟑ) are vertices of a triangle, Then its Incenter is the point of intersection of the angle bisectors. It is given by—

풙ퟎ =풂풙ퟏ + 풃풙ퟐ + 풄풙ퟑ

풂 + 풃 + 풄

풚ퟎ =풂풚ퟏ + 풃풚ퟐ + 풄풚ퟑ

풂 + 풃 + 풄

1.1.8 Circum-center— Let 퐀(풙ퟏ,풚ퟏ), 퐁(풙ퟐ,풚ퟐ) and 퐂(풙ퟑ,풚ퟑ) are vertices of a triangle, Then its Circum-center is the point of intersection of the side perpendicular bisectors. It is given by—

풙ퟎ =

풙ퟏퟐ 풚ퟏ

ퟐ 풚ퟏ ퟏ풙ퟐퟐ 풚ퟐ

ퟐ 풚ퟐ ퟏ풙ퟑퟐ 풚ퟑ

ퟐ 풚ퟑ ퟏ

ퟐ풙ퟏ 풚ퟏ ퟏ풙ퟐ 풚ퟐ ퟏ풙ퟑ 풚ퟑ ퟏ

풚ퟎ =

풙ퟏ 풙ퟏퟐ 풚ퟏ

ퟐ ퟏ풙ퟐ 풙ퟐ

ퟐ 풚ퟐퟐ ퟏ

풙ퟑ 풙ퟑퟐ 풚ퟑ

ퟐ ퟏ

ퟐ풙ퟏ 풚ퟏ ퟏ풙ퟐ 풚ퟐ ퟏ풙ퟑ 풚ퟑ ퟏ

Centroid

Incenter

∆=12

푥 푥 푥푦 푦 푦1 1 1

∆=12

|푥 (푦 − 푦 ) + 푥 (푦 − 푦 ) + 푥 (푦 − 푦 )|

Note—These three points will be collinear if ∆= ퟎ.

Circum-center

Unit-I COORDINATE GEOMETRY

RAI UNIVERSITY, AHMEDABAD

1.1.9 Orthocenter— Let 퐀(풙ퟏ,풚ퟏ), 퐁(풙ퟐ,풚ퟐ) and 퐂(풙ퟑ,풚ퟑ) are vertices of a triangle, Then its Orthocenter is the point of intersection of the altitudes of the triangle. It is given by—

풙ퟎ =

풚ퟏ 풙ퟐ풙ퟑ 풚ퟏퟐ ퟏ

풚ퟐ 풙ퟑ풙ퟏ 풚ퟐퟐ ퟏ

풚ퟑ 풙ퟏ풙ퟐ 풚ퟑퟐ ퟏ

풙ퟏ 풚ퟏ ퟏ풙ퟐ 풚ퟐ ퟏ풙ퟑ 풚ퟑ ퟏ

풚ퟎ =

풙ퟏퟐ 풚ퟐ풚ퟑ 풙ퟏ ퟏ풙ퟐퟐ 풚ퟑ풚ퟏ 풙ퟐ ퟏ풙ퟑퟐ 풚ퟏ풚ퟐ 풙ퟑ ퟏ풙ퟏ 풚ퟏ ퟏ풙ퟐ 풚ퟐ ퟏ풙ퟑ 풚ퟑ ퟏ

1.1.10 The Euler Line—The Euler line of a triangle is the line which passes through the orthocenter, circum-center, and centroid of the triangle.

Orthocenter

Unit-I COORDINATE GEOMETRY

RAI UNIVERSITY, AHMEDABAD

EXERCISE-I Question— A point divides internally the line- segment joining the points (8, 9) and (-7, 4) in the ratio 2 : 3. Find the co-ordinates of the point. Question— A (4, 5) and B (7, - 1) are two given points and the point C divides the line segment AB externally in the ratio 4 : 3. Find the co-ordinates of C. Question— Find the ratio in which the line-segment joining the points (5, - 4) and (2, 3) is divided by the x-axis. Also find the mid-point of these two points. Question—Find the centroid, in centre, circumcentre and orthocenter of a triangle whose vertices are (5, 3), (6, 1) and (7, 8). Also find the area of the triangle.

Unit-I COORDINATE GEOMETRY

RAI UNIVERSITY, AHMEDABAD

1.2.1 Straight line—The shortest path of line joining of any two point gives rise to a straight line. A general equation of a straight line is given by— 퐚풙 + 퐛풚 + 퐜 = ퟎ.

1.2.2 Slope of a straight line—Slope of any straight line is the tangent of angle (−휋 < 휃 ≤ 휋) make by the positive 푥 −axis. Slope of a line is also called as gradient and it is denoted by m. Slope of the line joining the points P(푥 ,푦 ) and Q(푥 ,푦 ) is given by— 1.2.3 Different form of equation of straight lines— Slope-intercept form—The equation of a straight line having slope m and an intercept c on 푦 −axis is

풚 = 풎풙 + 풄. Slope-Point form— The equation of a straight line passing through the fixed point(푥 ,푦 ) and having slope m is

풚 − 풚ퟏ = 풎(풙 − 풙ퟏ) Two point form—The equation of the line passing through two fixed points A(푥 ,푦 ) and B(푥 ,푦 ) is

풚 − 풚ퟏ =풚ퟐ − 풚ퟏ풙ퟐ − 풙ퟏ

(풙 − 풙ퟏ)

Intercept form—The equation of the line cutting off intercepts a and b on 푥 −axis and 푦 −axis respectively is

풙풂

+ 풚풃

= ퟏ Parametric form— The equation of a straight line passing through a fixed point P(x , y ) and making an angle θ, 0 ≤ θ ≤ π with positive direction of x −axis is—

풙 − 풙ퟏ퐜퐨퐬휽 =

풚 − 풚ퟏ퐬퐢퐧휽

풎 = 퐭퐚퐧휽 = 풚ퟐ 풚ퟏ

풙ퟐ 풙ퟏ

Unit-I COORDINATE GEOMETRY

RAI UNIVERSITY, AHMEDABAD

Normal or perpendicular form—

1.2.4 Angle between two straight lines— The angle between two straight lines is given by—

Condition for coincident, parallel, perpendicular and intersecting of two straight lines—

1. coincident— Two lines 푎 푥 + 푏 푦 + 푐 = 0 and 푎 푥 + 푏 푦 + 푐 = 0 will be coincident if— 푎푎 =

푏푏 =

푐푐

2. Parallel—Two straight lines are said to be parallel if their slope will be equal. (a) Lines 푎 푥 + 푏 푦 + 푐 = 0 and 푎 푥 + 푏 푦 + 푐 = 0 will be parallel if—

푎푎 =

푏푏 ≠

푐푐

(b) Lines 푦 = 푚 푥 + 푐 and 푦 = 푚 푥 + 푐 will be parallel if— 푚 = 푚

3. Perpendicular— Two straight lines are said to be perpendicular if the product of their slopes will be equal to −1. (a) The lines a x + b y + c = 0 and a x + b y + c = 0 will be perpendicular if—

푎 푎 + 푏 푏 = 0 (b) Lines 푦 = 푚 푥 + 푐 and 푦 = 푚 푥 + 푐 will be perpendicular if—

푚 푚 = −1 4. Intersecting— Two lines 푎 푥 + 푏 푦 + 푐 = 0 and 푎 푥 + 푏 푦 + 푐 = 0 intersects each other if—

푎푎 ≠

푏푏

Length of perpendicular— Length of perpendicular drawn from a point P(푥 ,푦 ) to a straight line 푎푥 + 푏푦 +푐 = 0 is—

푝 =|푎푥 + 푏푦 + 푐|

√푎 + 푏

Distance between two parallel lines— Distance between two parallel lines 푎푥 + 푏푦 + 푐 = 0 and 푎푥 + 푏푦 +푐 = 0 is—

푝 =|푐 − 푐 |

√푎 + 푏

The equation of a straight line in terms of the length of perpendicular p from the origin upon it and the angle 훼 which this perpendicular makes with the positive direction of 푥 −axis is— 풙 풄풐풔 + 풚 풔풊풏 = 풑

퐭퐚퐧 휽 =풎ퟏ −풎ퟐ

ퟏ + 풎ퟏ풎ퟐ=

풂ퟏ풃ퟐ − 풂ퟐ풃ퟏ풂ퟏ풂ퟐ + 풃ퟏ풃ퟐ

Unit-I COORDINATE GEOMETRY

RAI UNIVERSITY, AHMEDABAD

EXERCISE-II Question—Find the equation of a straight line which makes an angle of 135° with the x −axis and cuts y −axis at a distance − 5 from the origin. Question—Find the equation of the straight line which passes through the point (1, - 2) and cuts off equal intercepts from axes. Question—Find the equation of a straight line passing through (-3 , 2) and cutting an intercept equal in magnitude but opposite in sign from the axes. Question—If the coordinates of A and B be (1, 1) and (5, 7), then find the equation of the perpendicular bisector of the line segment AB. Question—Find the equation of the straight line passing through the point(1,2) and having a slope of 3. Question—Find the equation of the straight line passing through the point(3,2) and parallel to the line 2x + 3y = 1. Question—Find the equation of the straight line passing through the point(4,1) and perpendicular to 3x + 4y + 2 = 0. Question— Find the equation of a line through the points (1,2) and (3,1). What is its slope? What is its y intercept?

Unit-I COORDINATE GEOMETRY

RAI UNIVERSITY, AHMEDABAD

1.3.1 Circle— A circle is the set of all points that are at the same distance r from a fixed point. A general equation of a circle is—

풙ퟐ + 풚ퟐ + ퟐ품풙 + ퟐ풇풚 + 풄 = ퟎ. This equation represents a circle having center at (−푔,−푓) and radius 푟 = 푔 + 푓 − 푐 There are several standard forms of equation of a circle as given below— Center-radius form—The equation of a circle having center(ℎ, 푘) and radius r is—

(풙 − 풉)ퟐ + (풚 − 풌)ퟐ = 풓ퟐ Center at origin—The equation of a circle having center at origin and radius r is—

풙ퟐ + 풚ퟐ = 풓ퟐ Diameter form—Let A(푥 ,푦 ) and B(푥 ,푦 ) are the extremities of a diameter of the circle. Then the equation of the circle is— (풙 − 풙ퟏ)(풙 − 풙ퟐ) + (풚 − 풚ퟏ)(풚 − 풚ퟐ) = ퟎ Three point form—The equation of the circle passing through the points A(푥 ,푦 ), B(푥 ,푦 ) and C(푥 ,푦 )is—

푥 + 푦 푥 푦 1푥 + 푦 푥 푦 1푥 + 푦 푥 푦 1푥 + 푦 푥 푦 1

= 0.

Polar form— The polar equation of a circle centered at P(r ,θ ) with radius R units −

r + r − 2rr cos( θ − θ ) = R

퐶(−푔,−푓)

r

Unit-I COORDINATE GEOMETRY

RAI UNIVERSITY, AHMEDABAD

1.3.2 Tangent to a circle—There are two types of tangent to any circle—

1. Tangent through a point on the circle 2. Tangent from a point outside to the circle

Tangent through a point on the Circle— Let P(푥 ,푦 ) is a point on the circle. The equation of the tangent line is as given below—

Equation of circle Equation of tangent line

(푥 − ℎ) + (푦 − 푘) = 푟 (푥 − ℎ)(푥 − ℎ) + (푦 − 푘)(푦 − 푘) = 푟

푥 + 푦 = 푟 푥푥 + 푦푦 = 푟

푥 + 푦 + 2푔푥 + 2푓푦 + 푐 = 0 푥푥 + 푦푦 + 푔(푥 + 푥 ) + 푓(푦 + 푦 ) + 푐 = 0

Tangent from a point 퐏(풙ퟏ,풚ퟏ) outside to the circle— If 푥 + 푦 + 2푔푥 + 2푓푦 + 푐 = 0 is the equation of the circle, then— S ≡ 푥 + 푦 + 2푔푥 + 2푓푦 + 푐 S ≡ 푥 + 푦 + 2푔푥 + 2푓푦 + 푐 T ≡ 푥푥 + 푦푦 + 푔(푥 + 푥 ) + 푓(푦 + 푦 ) + 푐

Normal line—A normal line always pass through the center of the circle. So, we can easily find out the equation of normal by using 2-points form.

A Pair of tangents lines from an external

point P(푥 ,푦 ) is 퐒퐒ퟏ = 퐓ퟐ The cord of contact with respect to external point

P(푥 ,푦 ) is 퐓 = ퟎ

Unit-I COORDINATE GEOMETRY

RAI UNIVERSITY, AHMEDABAD

EXERCISE-III

Question—Find the center of the circle 푥 + 푦 + 20푥 − 10푦 − 19 = 0. Question—Find the area of the circle 4푥 + 4푦 + 16푥 − 16푦 − 16 = 0. Question—Find the equation of the circle having center at origin and radius 2. Question—Find the equation of the circle having center(2,3) and radius 4. Question—Find the equation of the circle having extremities of diameter (2,5) and (4,1). Question—Find the equation of the circle Passing through the points (1,3), (3,−1) and (2,4). Question—Find the equation of the tangent line to the circle x + y + 4x − 5y + 1 = 0 at the point(1,2). Question—Find the equation of the tangent line to the circle 푥 + 푦 − 2푥 − 4푦 + 6 = 0 at the point(1,2). Question—Find the equation of the normal line to the circle 푥 + 푦 + 2푥 − 4푦 = 0 at the point(1,1). Question—Find the radius of the circle having tangents 2푥 + 3푦 + 6 = 0 and 2푥 + 3푦 − 3 = 0.

Unit-I COORDINATE GEOMETRY

RAI UNIVERSITY, AHMEDABAD

Reference— 1. en.wikipedia.org/wiki/Analytic_geometry 2. https://www.khanacademy.org 3. www.stewartcalculus.com 4. www.britannica.com 5. mathworld.wolfram.com 6. www.mathsisfun.com 7. www.purplemath.com 8. http://www.education.gov.za/ 9. http://www.cut-the-knot.org/ 10. math2.org

Unit-I COORDINATE GEOMETRY

RAI UNIVERSITY, AHMEDABAD