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Basic Coordinate Geometry- Session for RankersBatch
Vidyalankar Institute
March 8, 2010
Basic Coordinate Geometry- Session for Rankers Batch
Motivation for Coordinate Geometry
Coordinate geometry is a mixture of Geometry and Algebra,A geometrical point is represented by a algebraic values (x , y). Thisgives us the power of both thinking procedures.For example : A line ax + by + c = 0 is an algebraic equationwhich satisfies all (x , y) which we call its solution set (in Algebra)but same can be interpreted as a point lying on a line
Basic Coordinate Geometry- Session for Rankers Batch
Transformations
Shift of origin
Invariant
Distance between two pointsSlope of a line
Rotation of axes about origin
Invariant
Distance between two pointsDistance between origin and a line
Rotation can be attained using multiplication by e iθ or
orthonormal matrix[
cos θ sin θ− sin θ cos θ
]to result rotation by
angle θ or a point (x , y)
Basic Coordinate Geometry- Session for Rankers Batch
Distance of point from a line
Distance of a point (x0, y0) from a line ax + by + c = 0 alonga direction θ is given by
AB =
∣∣∣∣ ax0 + by0 + ca cos θ + b sin θ
∣∣∣∣Shortest distance of (x0, y0) from a line ax + by + c = 0 is
AM =|ax0 + by0 + c |√
a2 + b2
Basic Coordinate Geometry- Session for Rankers Batch
Foot of the perpendicular & Image of a point
Foot of the perpendicular from (x0, y0) on the lineax + by + c = 0
x − x0
a=
y − y0
b= −
(ax0 + by0 + c
a2 + b2
)
Image of the point (x0, y0) in the line ax + by + c = 0
x − x0
a=
y − y0
b= −2
(ax0 + by0 + c
a2 + b2
)
Basic Coordinate Geometry- Session for Rankers Batch
Linear Combination of lines u=0,v=0
Linear combination of lines u = 0, v = 0 is
αu + βv = 0 or u + kv = 0
where α, β ∈ R1
Linear combination αu + βv = 0 always passes through theintersection of the constituent lines u = 0 and v = 0
1In linear combination, u & v can represent any geometrical structure.Basic Coordinate Geometry- Session for Rankers Batch
Two lines are parallel
Two lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 areparallel iff
a1
a2=
b1
b26= c1
c2
Special case : If these lines are coincident then
a1
a2=
b1
b2=
c1
c2
ExampleProblem solving : Equation of a line parallel to 3x + 2y + 5 = 0 is3x + 2y + c = 0 where c is identifiable using another condition.
Basic Coordinate Geometry- Session for Rankers Batch
Two lines are non-parallel
If two lines are non-parallel then they intersect in a point, withacute and obtuse angle between them.
tan θ =
∣∣∣∣ m1 −m2
1 + m1m2
∣∣∣∣where θ is acute angle between them
If they are coincident then θ = 0 and hence m1 = m2If they are perpendicular2 then θ = 0 and hence m1m2 = −1
ExampleEquation of line perpendicular to line ax + by + c = 0 isbx − ay + c2 = 0. Just interchange the coefficients a, b and replaceone sign negative
2Slope of line perpendicular to a line of slope m1 is −1/m1
Basic Coordinate Geometry- Session for Rankers Batch
Three lines are concurrent
Three lines aix + biy + ci = 0 are concurrent iff
∆ =
∣∣∣∣∣∣a1 b1 c1a2 b2 c2a3 b3 c3
∣∣∣∣∣∣ = 0
If they are not concurrent then they form a triangle and areaof this triangle is
A =12
∆2
C1C2C3
where Ci is the cofactor of ci term
Basic Coordinate Geometry- Session for Rankers Batch
Angle Bisector, Origin, Acute angle bisector
Equation of Angle bisector
a1x + b1y + c1√a21 + b2
1
= ±a2x + b2y + c2√a22 + b2
2
Equation of angle bisector containing origin (providedc1, c2 > 0)
a1x + b1y + c1√a21 + b2
1
=a2x + b2y + c2√
a22 + b2
2
If a1a2 + b1b2 > 0 ⇔ Obtuse angle contains the originIf a1a2 + b1b2 < 0 ⇔ Acute angle contains the origin
Basic Coordinate Geometry- Session for Rankers Batch
Locus
DefinitionLocus is defined as path traced by a variable point moving underconstraints. Mathematically, the path traced is the relationship between thex and y coordinate of any instance taken by the locus point (variable point).
Steps to solve problems of locus1 Let the locus point be (h, k)(we call this incoming variables)
[this gives an instance for the moving point]2 Introduce some new variables (we call them outgoing
variables) so that we proceed in the solution3 Find a relationship (geometrical) between the introduced
variable(s)4 Find relation between the incoming variables (h, k) and
outgoing variables5 Replace (h, k) by (x , y)
Basic Coordinate Geometry- Session for Rankers Batch
Two lines - Angle bisector/Origin
ExampleTwo lines y = x + 1 and y + 1 = 0 are given then
find equation of angle bisector containing the origin
find obtuse angle between them?
where does the origin lie in (acute or obtuse angle)?
find obtuse angle bisector
Basic Coordinate Geometry- Session for Rankers Batch
Two lines - Angle bisector/Origin
ExampleTwo lines y = x + 1 and y + 1 = 0 are given then
find equation of angle bisector containing the origin
find obtuse angle between them?
where does the origin lie in (acute or obtuse angle)?
find obtuse angle bisector
Basic Coordinate Geometry- Session for Rankers Batch
Two lines - Angle bisector/Origin
ExampleTwo lines y = x + 1 and y + 1 = 0 are given then
find equation of angle bisector containing the origin
find obtuse angle between them?
where does the origin lie in (acute or obtuse angle)?
find obtuse angle bisector
Basic Coordinate Geometry- Session for Rankers Batch
Two lines - Angle bisector/Origin
ExampleTwo lines y = x + 1 and y + 1 = 0 are given then
find equation of angle bisector containing the origin
find obtuse angle between them?
where does the origin lie in (acute or obtuse angle)?
find obtuse angle bisector
Basic Coordinate Geometry- Session for Rankers Batch
Three lines
ExampleGiven three lines 2x + y − 3 = 0, x + 4y − 5 = 0 and3x + 5y − 1 = 0. Find the area bounded by these three lines
ExampleFind the value of c such that 2x + y − 3 = 0, x + 4y − 5 = 0 and3x + 5y + c = 0 are concurrent using methods mentioned below
Using determinant approach
Using linear combination of the form u + kv = 0 andαu + βv = 0
Try this scenario using both kinds of linear combination forms2x + y − 3 = 0, x + 4y − 5 = 0 and ax + 14y − 21 = 0
Basic Coordinate Geometry- Session for Rankers Batch
Three lines
ExampleGiven three lines 2x + y − 3 = 0, x + 4y − 5 = 0 and3x + 5y − 1 = 0. Find the area bounded by these three lines
ExampleFind the value of c such that 2x + y − 3 = 0, x + 4y − 5 = 0 and3x + 5y + c = 0 are concurrent using methods mentioned below
Using determinant approach
Using linear combination of the form u + kv = 0 andαu + βv = 0
Try this scenario using both kinds of linear combination forms2x + y − 3 = 0, x + 4y − 5 = 0 and ax + 14y − 21 = 0
Basic Coordinate Geometry- Session for Rankers Batch
Three lines
ExampleGiven three lines 2x + y − 3 = 0, x + 4y − 5 = 0 and3x + 5y − 1 = 0. Find the area bounded by these three lines
ExampleFind the value of c such that 2x + y − 3 = 0, x + 4y − 5 = 0 and3x + 5y + c = 0 are concurrent using methods mentioned below
Using determinant approach
Using linear combination of the form u + kv = 0 andαu + βv = 0
Try this scenario using both kinds of linear combination forms2x + y − 3 = 0, x + 4y − 5 = 0 and ax + 14y − 21 = 0
Basic Coordinate Geometry- Session for Rankers Batch
Three lines
ExampleGiven three lines 2x + y − 3 = 0, x + 4y − 5 = 0 and3x + 5y − 1 = 0. Find the area bounded by these three lines
ExampleFind the value of c such that 2x + y − 3 = 0, x + 4y − 5 = 0 and3x + 5y + c = 0 are concurrent using methods mentioned below
Using determinant approach
Using linear combination of the form u + kv = 0 andαu + βv = 0
Try this scenario using both kinds of linear combination forms2x + y − 3 = 0, x + 4y − 5 = 0 and ax + 14y − 21 = 0
Basic Coordinate Geometry- Session for Rankers Batch
Point and a line
ExampleThe equations of the perpendicular bisectors of the sides AB andAC of triangle ABC are x − y + 5 = 0 and x + 2y = 0 respectively.If the point A is (1,−2) find the equation of the line .
Example
If a point P(a2, a) lies in the region corresponding to the obtuseangle between y + 2 = 0 and y = x + 1 then find the intervalwhere a belongs to?
Example
If a point P(a2, a) lies in the region corresponding to the acuteangle between the lines 2y = x and 4y = x then find the intervalwhere a belongs to?
Basic Coordinate Geometry- Session for Rankers Batch
Point and a line
ExampleThe equations of the perpendicular bisectors of the sides AB andAC of triangle ABC are x − y + 5 = 0 and x + 2y = 0 respectively.If the point A is (1,−2) find the equation of the line .
Example
If a point P(a2, a) lies in the region corresponding to the obtuseangle between y + 2 = 0 and y = x + 1 then find the intervalwhere a belongs to?
Example
If a point P(a2, a) lies in the region corresponding to the acuteangle between the lines 2y = x and 4y = x then find the intervalwhere a belongs to?
Basic Coordinate Geometry- Session for Rankers Batch
Point and a line
ExampleThe equations of the perpendicular bisectors of the sides AB andAC of triangle ABC are x − y + 5 = 0 and x + 2y = 0 respectively.If the point A is (1,−2) find the equation of the line .
Example
If a point P(a2, a) lies in the region corresponding to the obtuseangle between y + 2 = 0 and y = x + 1 then find the intervalwhere a belongs to?
Example
If a point P(a2, a) lies in the region corresponding to the acuteangle between the lines 2y = x and 4y = x then find the intervalwhere a belongs to?
Basic Coordinate Geometry- Session for Rankers Batch
Point & Line continued...
Problem :A ray of light is sent along the line 2x − 3y = 5. After refractingacross the line x + y = 1 it enters the opposite side after turning by15◦away from the line x + y = 1. FInd the equation of the linealong which the refracted ray travels.
Diagram
Basic Coordinate Geometry- Session for Rankers Batch
Point & Line continued...
Problem :A ray of light is sent along the line 2x − 3y = 5. After refractingacross the line x + y = 1 it enters the opposite side after turning by15◦away from the line x + y = 1. FInd the equation of the linealong which the refracted ray travels.
Diagram
Basic Coordinate Geometry- Session for Rankers Batch
Point & Line continued...
Example
Find the point (x , y) on 2x + 3y = 6 which is atShortest distance from (3, 2)
Distance of 2 from (3, 2)
Basic Coordinate Geometry- Session for Rankers Batch
Point & Line continued...
Example
Find the point (x , y) on 2x + 3y = 6 which is atShortest distance from (3, 2)
Distance of 2 from (3, 2)
Basic Coordinate Geometry- Session for Rankers Batch
Problems on Locus
Example
If the coordinates of a variable point P be(
t +1t, t − 1
t
)where t
is a variable quantity, then find the locus of the point P
ExampleThe variable line x cos θ + y sin θ = 2 cuts the X and Y axis at Aand B respectively. Find the locus of vertex of point P of therectangle OAPB where O is the origin.
ExampleA variable line is at a constant distance p from the origin andmeets the coordinate axes in A and B. Show that locus of thecentroid of ∆OAB is x−2 + y−2 = 9p−2
Basic Coordinate Geometry- Session for Rankers Batch
Problems on Locus
Example
If the coordinates of a variable point P be(
t +1t, t − 1
t
)where t
is a variable quantity, then find the locus of the point P
ExampleThe variable line x cos θ + y sin θ = 2 cuts the X and Y axis at Aand B respectively. Find the locus of vertex of point P of therectangle OAPB where O is the origin.
ExampleA variable line is at a constant distance p from the origin andmeets the coordinate axes in A and B. Show that locus of thecentroid of ∆OAB is x−2 + y−2 = 9p−2
Basic Coordinate Geometry- Session for Rankers Batch
Problems on Locus
Example
If the coordinates of a variable point P be(
t +1t, t − 1
t
)where t
is a variable quantity, then find the locus of the point P
ExampleThe variable line x cos θ + y sin θ = 2 cuts the X and Y axis at Aand B respectively. Find the locus of vertex of point P of therectangle OAPB where O is the origin.
ExampleA variable line is at a constant distance p from the origin andmeets the coordinate axes in A and B. Show that locus of thecentroid of ∆OAB is x−2 + y−2 = 9p−2
Basic Coordinate Geometry- Session for Rankers Batch
Problems on Locus
Example
P is any pt on the line x − a = 0. If A is the point (a, 0) andPQ,the bisector of the angle OPA, meets the x-axis in Q. Provethat the locus of the foot of the perpendicular from Q on OP is
(x − a)2(x2 + y2) = a2y2
Basic Coordinate Geometry- Session for Rankers Batch
Problems on Locus
Example
P is any pt on the line x − a = 0. If A is the point (a, 0) andPQ,the bisector of the angle OPA, meets the x-axis in Q. Provethat the locus of the foot of the perpendicular from Q on OP is
(x − a)2(x2 + y2) = a2y2
Basic Coordinate Geometry- Session for Rankers Batch
Problems on Locus
Example
A quadratic equation f (x) = ax2 + bx + c = 0 where a, b, c ∈ R isgiven. For a given value of a and c we vary b such that the locus ofvertex is y = g(x) then
1 Is a line parallel to y − axis2 Is parallel to x − axis3 Is a quadratic curve4 Is a point
Basic Coordinate Geometry- Session for Rankers Batch
Maxima-Minima Problems
Example
A man starts from P(−3, 4) and reaches Q(0, 1) touching X − axisat R(α, 0) such that PR + RQ is minimum then α =?
Example
Consider the points A(0, 1) and B(2, 0), P is a point on the linex + y + 1 = 0. Find coordinates of point P such that |PA− PB| is
MaximumMinimum
Also find the max and min value attained by |PA− PB|
Basic Coordinate Geometry- Session for Rankers Batch
Maxima-Minima Problems
Example
A man starts from P(−3, 4) and reaches Q(0, 1) touching X − axisat R(α, 0) such that PR + RQ is minimum then α =?
Example
Consider the points A(0, 1) and B(2, 0), P is a point on the linex + y + 1 = 0. Find coordinates of point P such that |PA− PB| is
MaximumMinimum
Also find the max and min value attained by |PA− PB|
Basic Coordinate Geometry- Session for Rankers Batch
Transformations
ExampleGiven a line 3x + 4y + 1 = 0 in coordinate system, on shifting theorigin to a new location (h, k) if the x − intercept of the line in thenew coordinate system is 2 then what is its y − intercept
ExampleIn the above problem, if instead of shifting the origin to a newlocation the system of axes is rotated about the origin by someangle θ with new x − intercept of 2 then what is the y − interceptmade by the line?
ExampleThrough what angle should the axes be rotated so that the equation9x2 − 2
√3xy + 7y2 = 10 may be changed to 3x2 + 5y2 = 5?
Basic Coordinate Geometry- Session for Rankers Batch
Transformations
ExampleGiven a line 3x + 4y + 1 = 0 in coordinate system, on shifting theorigin to a new location (h, k) if the x − intercept of the line in thenew coordinate system is 2 then what is its y − intercept
ExampleIn the above problem, if instead of shifting the origin to a newlocation the system of axes is rotated about the origin by someangle θ with new x − intercept of 2 then what is the y − interceptmade by the line?
ExampleThrough what angle should the axes be rotated so that the equation9x2 − 2
√3xy + 7y2 = 10 may be changed to 3x2 + 5y2 = 5?
Basic Coordinate Geometry- Session for Rankers Batch
Transformations
ExampleGiven a line 3x + 4y + 1 = 0 in coordinate system, on shifting theorigin to a new location (h, k) if the x − intercept of the line in thenew coordinate system is 2 then what is its y − intercept
ExampleIn the above problem, if instead of shifting the origin to a newlocation the system of axes is rotated about the origin by someangle θ with new x − intercept of 2 then what is the y − interceptmade by the line?
ExampleThrough what angle should the axes be rotated so that the equation9x2 − 2
√3xy + 7y2 = 10 may be changed to 3x2 + 5y2 = 5?
Basic Coordinate Geometry- Session for Rankers Batch
Linear Combination
Example
For a family of lines 5x + 3y − 2 + λ(3x − y − 4) = 0 andx − y + 1 + µ(2x − y − 2) = 0 then
Equation of the straight line that belongs to both the families
Locus of the point of intersection of lines from each familyintersecting at right angles to each other. Hence also derivethe equation of a circle given coordinates of the end points ofits diameter.
Example
If a2 + c2 − b2 + 2ac = 0 then the family of straight linesax + by + c = 0 is concurrent at the points?
Basic Coordinate Geometry- Session for Rankers Batch
Linear Combination
Example
For a family of lines 5x + 3y − 2 + λ(3x − y − 4) = 0 andx − y + 1 + µ(2x − y − 2) = 0 then
Equation of the straight line that belongs to both the families
Locus of the point of intersection of lines from each familyintersecting at right angles to each other. Hence also derivethe equation of a circle given coordinates of the end points ofits diameter.
Example
If a2 + c2 − b2 + 2ac = 0 then the family of straight linesax + by + c = 0 is concurrent at the points?
Basic Coordinate Geometry- Session for Rankers Batch
Linear Combination
Example
For a family of lines 5x + 3y − 2 + λ(3x − y − 4) = 0 andx − y + 1 + µ(2x − y − 2) = 0 then
Equation of the straight line that belongs to both the families
Locus of the point of intersection of lines from each familyintersecting at right angles to each other. Hence also derivethe equation of a circle given coordinates of the end points ofits diameter.
Example
If a2 + c2 − b2 + 2ac = 0 then the family of straight linesax + by + c = 0 is concurrent at the points?
Basic Coordinate Geometry- Session for Rankers Batch
Linear combination continued...
Example
Given the family of lines a(3x + 4y + 6) + b(x + y + 2) = 0. Theline of the family situated at the greatest distance from the pointP(2, 3) has equationA) 4x + 3y + 8 = 0 B) 5x + 3y + 10 = 0 C) 15x + 8y + 30 = 0 D)None
Example
If 6a2 − 3b2 − c2 + 7ab − ac + 4bc = 0 then the family of linesax + by + c = 0 is concurrent atA) (−2,−3) B) (3,−1) C) (2, 3) D) (−3, 1)
Basic Coordinate Geometry- Session for Rankers Batch
Linear combination continued...
Example
Given the family of lines a(3x + 4y + 6) + b(x + y + 2) = 0. Theline of the family situated at the greatest distance from the pointP(2, 3) has equationA) 4x + 3y + 8 = 0 B) 5x + 3y + 10 = 0 C) 15x + 8y + 30 = 0 D)None
Example
If 6a2 − 3b2 − c2 + 7ab − ac + 4bc = 0 then the family of linesax + by + c = 0 is concurrent atA) (−2,−3) B) (3,−1) C) (2, 3) D) (−3, 1)
Basic Coordinate Geometry- Session for Rankers Batch
Miscellaneous Problems
Problem #1 :
If4∑
i=1
(x2i + y2
i ) ≤ 2x1x3 + 2x2x4 + 2y2y3 + 2y1y4
then the points (xi , yi ) for i = 1 to 4 areA) vertices of a rectangleB) CollinearC) ConcyclicD) None of these
Basic Coordinate Geometry- Session for Rankers Batch
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