Complete Coordinate Geometry
Complete Coordinate Geometry
Complete Coordinate Geometry
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Q (x2, y2)
P (x1, y1)
Section Formula
B. 2/5
C. 2/√5
D. √3/5
Important Points w.r.t Triangle
Important Points w.r.t. Triangle
Point of Intersection of
Centroid Medians
Orthocenter Altitudes
B. 1 + 2 √5
C. 2 +√5
D. 2 √5 - 1
Concurrency of 3 Lines
Concurrency Of Three Lines
Three lines are concurrent if
The number of values of p for which the lines x + y - 1 = 0, px + 2y + 1 = 0
and 4x + 2py + 7 = 0 are concurrent is equal to
A. 0
B. 2
C. 1
D. infinite
Family of Lines
Family Of Straight Lines
Equation of line passing through the intersection of L1 & L2 is given by L1 +λ L2 =0
Example
If the family of straight lines x(a + 2b) + y(a + 3b) = a + b passes
through a fixed point for all values of a and b. Find the point.
The equation of the locus of the foot of perpendicular drawn from
(5,6) on the family of lines (x - 2) + λ(y - 3) = 0
NTA Question
A. (x - 1)(x - 3) + (y - 2)(y - 6) = 0
B. (x - 5)(x - 6) + (y - 2) (y - 3) = 0
C. (x - 2)(x - 5) + (y - 3) (y - 6) = 0
D. (x + 2) (x + 5) + (y + 3) (y + 6) = 0
Optics Based Questions
Optics Based Problem
B. (x – 4)2 + (y + 2)2 = 16
C. (x – 4)2 + (y - 4)2 = 8
D. (x - 2)2 + (y - 2)2 = 12
Equation of Angle Bisector
Example
The equation of the bisector of the lines 3x - 4y + 1 = 0 and 5x + 12y - 11
= 0 which do not contain origin is
A. 7x - 56y + 34 = 0
B. 32x + 4y - 21=0
C. 3x + 7y + 11 = 0
D. 16x - 3y + 7 = 0
Example
The equation of the bisector of the acute angle between the lines 3x -
4y + 7 = 0 and 12x + 5y - 2 = 0 is
B. 11x - 3y + 9 = 0
D. 11x - 3y - 9 = 0
Circles
General Equation of Circle
C(a, b)
Intercepts made by a circle on the axes :
PYQ
If the area of an equilateral triangle inscribed in the circle,
x2 + y2 + 10x + 12y + c = 0 is 27√3 sq. units then c is equal to :
A. 13
D. 25
Parametric coordinates of endpoints of
diameter
(r cos𝜃, r sin𝜃)
C (0, 0)
(- r cos𝜃, - r sin𝜃)
Let PQ be a diameter of the circle x2 + y2 = 9. If α and β are the lengths of the
perpendiculars from P and Q on the straight line, x + y = 2 respectively, then the
maximum value of αβ is _
B. 5√3
C. 5√4
D. 4√5
The tangent and the normal lines at the point (√3, 1) to the circle x2 + y2 = 4 and
the x - axis form a triangle. The area of this triangle (in square units) is :
C. 2 / √3
D. 1 / √3
Length of Tangent
Length of tangent from external point
P (x1, y1)
Important Figure
O
P (x1, y1)
B
Important Note:
O
P
B
Let the tangents drawn from the origin to the circle,
x2 + y2 - 8x - 4y + 16 = 0 touch it at the points A and B. The (AB)2 is equal to :
B. 56/5
x2 + y2 - 8x - 4y + 16 = 0
C. 64/5
D. 32/5
P (0, 0)
Number of Common Tangents
Common tangents to two circles :
Common tangents to two circles :
PYQ
The minimum distance between any two points P1 and P2 while
considering point P1 on one circle and point P2 on the other circle for
the given circles equations
x2 + y2 - 10x - 10y + 41 = 0
x2 + y2 - 24x - 10y + 160 = 0 is __.
B. 6
C. 9
D. 4
Family of Circles
The line x = y touches a circle at the point (1, 1). If the circle also passes through
the point (1, -3), then its radius is :
B. 2√2
C. 2
D. 3√2
Parabola
Example :
The area (in sq. units) of an equilateral triangle inscribed in the
parabola y2 = 8x, with one of its vertices on the vertex of this
parabola, is :
A. 64 √3
C. 192 √3
D. 128 √3
The locus of a point which divides the line segment joining the point (0, -1) and a
point on the parabola, x2 = 4y, internally in the ratio 1 : 2 is :
A. 9x2 - 12y = 8
JEE Main 2020
B. 9x2 - 3y = 2
C. x2 - 3y = 2
D. 4x2 - 3y = 2
Length of Focal Chord
Length of focal chord
(a, 0)
If one end of a focal chord of the parabola, y2 = 16x is at (1, 4), then the length of
this focal chord is :
B. 22
C. 24
D. 20
Equation of Tangent to Parabola
Remember:
The equation of a tangent to the parabola, x2 = 8y, which makes an angle θ with
the positive direction of x-axis, is :
B. y = x tan θ - 2 cot θ
C. x = y cot θ + 2 tan θ
D. x = y cot - 2 tan θ
If one end of a focal chord AB of the parabola y2 = 8x is at A(1/2 , -2), then the
equation of the tangent to it at B is :
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A. 2x + y - 24 = 0
B. x - 2y + 8 = 0
C. x + 2y + 8 = 0
D. 2x - y - 24 = 0
Point of Intersection of Tangents
G O A
Common Tangents
Example :
If the common tangent to the parabolas, y2 = 4x and x2 = 4y also
touches the circle, x2 + y2 = c2, then c is equal to :
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A. 1 / 2√2
B. 1 / √2
C. 1/4
D. 1/2
Shortest Distance
If P is a point on the parabola y = x2 + 4 which is closest to the straight line
y = 4x - 1, then the coordinates of P are :
A. (-2, 8)
B. (1, 5)
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C. (3, 13)
D. (2, 8)
Standard Equation of Ellipse
If the distance between the foci of an ellipse is 6 and the distance between its
directrices is 12, then the length of its latus rectum is :
A. √3
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B. 3√2
C. 3/√2
D. 2√3
Equation of Tangent
Parametric Form
Slope Form
Let L be a tangent line to the parabola y2 = 4x - 20 at (6, 2). If L is also a
tangent to the ellipse then the value of b is equal to :
B. 14
C. 16
D. 20
Eqn of Normal :
1. Point Form
2. Parametric Form
3. Slope Form
Let x = 4 be a directrix to an ellipse whose centre is at the origin and its
Eccentricity is 1/2 . If P (1, β), β > 0 is a point on this ellipse, then the
equation of the normal to it at P is :
B. 8x - 2y = 5
C. 7x - 4y = 1
D. 4x - 2y = 1
Standard Equation of Hyperbola
A hyperbola passes through the foci of the ellipse x2/25 + y2/16 = 1 and
its transverse and conjugate axes coincide with major and minor axes
of the ellipse, respectively. If the product of their eccentricities is one,
then the equation of the hyperbola is
JEE Main 2021
A. x2/9 - y2/4 = 1
B. x2/9 - y2/16 = 1
C. x2 - y2 = 9
D. x2/9 - y2/25 = 1
Equation of Tangent :
Parametric Form
Slope Form
Equation of Normal:
Point Form
Parametric Form
Slope Form
If a hyperbola passes through the point P(10, 16) and it has vertices at (
士6, 0), then the equation of the normal to it at P is :
B. 2x + 5y = 100
C. x + 2y = 42
D. x + 3y = 58
If the focus of a hyperbola is (土3, 0) and the equation of a tangent is
2x + y - 4 = 0, then the equation of the hyperbola is
NTA Question
A. 4x2 - 5y2 = 20
B. 5x2 - 4y2 = 20
C. 4x2 - 5y2 = 1
D. 5x2 - 4y2 = 1
Rectangular Hyperbola xy=c2
Rectangular Hyperbola xy=c2 :
Kundli of Hyperbola xy=c2 :
Rectangular Hyperbola
Standard Equation xy = c2
Centre (0, 0)
Equation of asymptotes
Vertexes
Foci
Eqn of Directrix
Kundli of Hyperbola xy=c2 :
Rectangular Hyperbola
Length of Transverse
axis
Length of Latus
rectum(LR)
Parametric Coordinates
Auxiliary Circle
Director Circle
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