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COORDINATE GE0METRY
(CIRCLE)
By:- Nishant Gupta
For any help contact:
9953168795, 9268789880
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Lista de exercícios
Tipologia: Exercícios
2013
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COORDINATE GE0METRY
(CIRCLE)
By:- Nishant Gupta
For any help contact:
Nishant Gupta, D-122, Prashant vihar, Rohini, Delhi-
BASIC
- Standard form of equation of a circle :
Equation of a circle whose centre is (h, k} and radius r units is (x – h)^2 + (y - k)^2 = r^2 If centre is (0, 0), then equation of circle is x^2 +y^2 = r^2
- General form of equation of a circle :
A second degree equation in two variables x & y represents a circle if (i) Coefficients of x^2 and y^2 are equal. (ii) Terms containing the product of xy is missing. General equation of a circle is x^2 + y^2 + 2gx + 2fy + c = 0.
Centre is ( - g, -f ) =( - 2
coeff of x , - 2
coeff. Of y )
Radius is g^2 f^2 c
Note : While finding centre & radius a circle, we must make sure that the coefficients of x^2 & y^2 are 1 each.
- Parametric equations of a circle : Parametric equations of a circle whose centre is (h,k) and radius r is
x = h + r cos θ , y = k + r sin θ, ( θ is parameter)
- Diametric form :
If A (x 1 ,y 1 ) & B( x 2 , y 2 ) be diameter of a circle then its equation is (x – x 1 ) (x – x 2 ) + (y – y 1 ) (y – y 2 ) = 0
- Intercepts on axes by x^2 + y^2 + 2gx + 2fy + c = 0
(a) On X – axis 2 g^2 c
(b) On Y – axis 2 f^2 c
ADDITIONAL
CIRCLE
Nishant Gupta, D-122, Prashant vihar, Rohini, Delhi-
If r 1 ,r 2 be radii & d be distance between their centers then angle is given by Cosθ = 12
2 2 2
2 1 2 r r
r r d
- ORTHOGONALITY CONDITION x^2 + y^2 + 2gi x + 2f (^) iy + ci= 0 where i = 1, 2
2 (g 1 g 2 + f 1 f 2 ) = c 1 + c 2 NOT IN SYLLABUS OF AIEEE
- Radical axis :- Locus of a point which moves s.t. tangents from it to two circleS are of same length.
Equation S 1 – S 2 = 0 (same as common chord)
- Radical centre o f S 1 = 0, S 2 = 0 and S 3 = 0. Find radical axes S 1 – S 2 = 0 & S 2 – S 3 = 0 Their point of intersection is radical centre.
- Limiting point; Circle with radius zero. If (0, 0) is one limiting point then other is
(^2 2) f (^2) g 2
fc , f g
gc
- Co-axial system :- If every pair of a family of circle have same radical axis.
We have may take its equation as x^2 + y^2 + 2gx + c = 0
Nishant Gupta, D-122, Prashant vihar, Rohini, Delhi-
- 2x^2 +2kxy + 2y^2 + (k-4)x +6y – 5 = 0 represents circle of radius (a) 3√2 (b) 2√ (c) 2√2 (d) N/T
- Equation of the circle centred at (1, 2) and passing through the point of intersection of the lines x+ 2y= 3 and 3x +y =4. (a) x^2 + y^2 -x-2y+1= (b) x^2 + y^2 - 2x-4y +4= (c ) x^2 + y^2 - 2x-4y -4= (c) N/T
- If circle concentric with the circle x^2 + y^2 + 4x- 2y + 4 = 0 and with three times its radius. (a) x^2 + y^2 - 4x-2y -4= (b) x^2 + y^2 + 4x-2y -1= (c) x^2 + y^2 + 4x-2y- 4= (d) N/T
- The equation of diameter of circle x^2 + y^2 -2x
- 4 y = 0 which passes through origin is (a) x +2y = 0 (b) x – 2y = 0 (c) 2x +y = 0 (d) 2x – y = 0
- If 4x – 3y- 7 = 0 and 8x – 6y -39 = 0 are common tangents to a circle, then radius is (a) 5/2 (b) 7/ (c) 5/4 (d) 3/
- Equation of the circle having diameters 2x – 3y = 5 and 3x – 4y = 7 and radius 8 is (a) x^2 + y^2 + 2x + 2y – 2 = 0 (b) x^2 + y^2 + 2x - 2y + 62 = 0 (c) x^2 + y^2 + 2x + 2y - 62 = 0 (d) none of these.
]
- Number of tangents drawn from (- 5,2 ) to the circle x^2 + y^2 – 14x + 2y - 25 = 0 are (a) 0 (b) 2 (c) 1 (d) N/T
- Tangent to x^2 + y^2 + 4x - 4y + 4 = 0 making equal intercepts on axes ,is (a) x+ y =2 (b) x+ y = (c) x+ y=2√2 (d) x+ y = 8
- m =? such that y = mx + 2 cuts x^2 + y^2 = 1 at distinct / coincident pts
(a) [ - , - 3 ] U [ 3 , ] (b) [ - 3 , 3 ]
(c) [ 3 , ] (d) None
- x + y tanθ = cos θ touches x^2 + y^2 = 4 for θ (a) π/4 (b) π / (c) π /2 (d) N/T
- Two circles each of radius 5, have common tangent at (1, 1) whose eqn. is 3x+4y-7= 0. Then their centers are (a) (3, 4) , ( -2,3) (b) (4, -3), (-2,5) (c) ( 4, 5), (-2-3) (d) N/T
- The length of the tangent drawn form any point on the circle x^2 + y^2 + 2gx + 2fy + = 0 to the circle x^2 + y^2 + 2gx + 2fy + = 0 (a) ( - ) (b) ( - ) (c) ( + ) (d) None.
- If circle x^2 + y^2 – 6x – 4y + 9 = 0 bisects circumference of the circle x^2 + y^2 – 6y + k = 0, then k equals (a) 2 (b) (c) 15 (d) 1
ASSIGNMENT
CIRCLE
Nishant Gupta, D-122, Prashant vihar, Rohini, Delhi-
- Chord of contact of tangents from P to x^2 + y (^2) = a (^2) always touches x (^2) + y 2 – 2ax = 0 , locus of P is (a) y 2 = a( a-2x) (b) x^2 + y 2 = ( x- a) 2 (c) x 2 = a( a-2y) (d) N/T
- The locus of the center of the circle x^2 + y^2 + 4x cos (^) - 2 y sin (^) - 10 = 0 is (a) an ellipse (b) a circle (c)a hyperbola (d) a parabola
- Locus of poles of x 2 y^2 a^2 & so that
polar is always touching x 2 y^2 b^2 is
(a) a 2 (x^2 y^2 )b^4 (b)b 2 (x^2 y^2 )a^4
(c) x 2 y^2 a^2 b^2 (d) None
- Distances of centers of x^2 + y^2 – 2 (^) ix = c^2 ( i= 1,2,3 ) from origin are in GP then lengths of tangents from any pt. of x^2 + y^2 = c^2 to these circles are in (a) AP (b) GP (c) HP (d) None
- If chord of contact of tangents drawn from a
pt. on (^) x 2 y^2 a^2 to (^) x 2 y^2 b^2 touches x 2 y^2 c^2 , then a, b, c are in (a) A.P (b) G.P (c) H.P (d) None
- If abscissae & ordinates of points P & Q are roots of x^2 + 2ax = b^2 & x^2 + 2px = q^2 resp circle with P Q as diameter is (a) x^2 + y^2 + 2ax + 2py – b^2 – q^2 = 0 (b) x^2 + y^2 – 2ax – 2py + b^2 + q^2 = 0 (c) x^2 + y^2 – 2ax–2py– b^2 – q^2 = 0 (d) None
- Circles x^2 + y^2 + x + y = 0 & x^2 + y^2 + x – y = 0 intersect at an angle of (a) /6 (b) / (c) /3 (d) N/T
- Image of x^2 + y^2 – 6x + 8 = 0 & y = x is
(a) x^2 + y^2 – 6y + 8 = 0 (b) x^2 + y^2 – 6y = 8 (c) x^2 + y^2 + 6x + 8 = 0 (d) N/T
- Greatest distance of point (10, 7) from circle x^2 + y^2 – 4x – 2y – 20 = 0 is
(a) 10 (b) 15 (c) 5 (d) N/T
- Two circles, each of radius 5, have a common tangent at (1, 1) whose equation is 3x + 4y – 7 = 0. Then their centres are (a) (4, -5), (-2, 3) (b) (4, -3), (-2, 5) (c) (4, 5) , (-2, -3) (d) N/T
- If x 2 y^2 4 & x 2 y^2 4 9 =0 have two common tangent then is
(a)
(b) 8
or 8
(c) 13 / 8 (d) None
- Two circles x^2 + y^2 = 6 and x^2 + y^2 – 6x + 8 = 0 are given. Then circle through their point of intersection and the point (1, 1) is (a) x^2 + y^2 – 6x + 4 = 0 (b) x^2 + y^2 – 3x + 1 = 0 (c) x^2 + y^2 – 4y + 2 = 0 (d) N/T
- Let PQ & RS be tangents at extremities of diameter PR of circle of radius r. If PQ & RS intersect at X on circumference of the circle , then 2r is
(a) PQ.RS (b) 2
PQ RS
(c) PQ RS
2 PQ.RS
(d) 2
PQ 2 RS^2
- Two circles of radii a and b ( a > b ) touch each other externally. Then the radius of circle which touches both of them externally and also their direct common tangent is
(a)
2 a b
ab
(b) √ab
(c) (a+b ) /2 (d) N/ T
- If tangents to 1 b
y 9
x 2
2 2 through ( 1 ,2√3 )
are at right angles then b is (a) 2 (b) 3 (c) 4 (d) 1