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Allyson Chastity Bates
6 years ago
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1 COORDINATE GEOMETRY LOCUS EXERCISE 1. The locus of P(x,y) such that its distance from A(0,0) is less than 5 units is x y 5 ) x y 10 x y 5 4) x y 0. The equation of the locus of the point whose distance from x-axis is twice its distance from the y- axis, is y =4x )4y =x y=3x 4)4x+y=0 3. The equation to the locus of points equidistant from the points (,, (,5) is x y 4 0 ) x y 1 0 x y 4 0 4) x y If the equation of the locus of a point equidistant from the points a 1,b 1 and a a x b b y c 0 then the value of c is a b a b ) a1 a b1 b a a b b 4) a 1 b1 a b 1 1 a,b is 5. A9,0, B 1,0 are two points. If P is a point such that PA:PB = 3 : 1, then the locus of P is x y 9 ) x y 9 0 x y 9 4) x y The locus of the moving point P, such that PA = 3PB where A(0,0), B(4, is 5x 5y 7x 54y 5 0 ) 5x 5y 7x 54y 5 0 3x 3y 70x 5y 5 0 4) None 7. Sum of the squares of the distance from a point to (c,0) and ( c,0) is 4c. It s locus is x y c 0 ) x y 4c x y c 4) x y c 8. A(,, B(1,5),C( 1,) are three points. If P is a point moves such that PA PB PC, then the locus of P is 10x 8y 9 0 ) 10x 8y x 8y 9 0 4) 10x 8y 9 0

2 9. A,3,B 1,1 are two points. If P is a point such that APB 90, then the locus of P is x y x 4y 1 0 ) x y x 4y 1 0 x y x 4y 1 0 4) x y x 4y The locus of P such that are of PAB is 1 square units where A= (, and B=( 4,5) is x 6xy 9y x 66y 3 0 ) x 6xy 9y x 66y 3 0 x 6xy 9y x 66y 3 0 4) x 6xy 9y x 66y ,0, A4,0,B0,6 are three points. If P is a point such that area of POB is twice the area of POA, then the locus of P is 4x 6y 0 ) 3x 4y 0 9x 16y 0 4) 4x 9y 0 1. A(,, B(, are two points. The equation to the locus of P such that PA+PB=8 is 16x 7y 64x 48 0 ) 16x 7y 64x x 7y 64x ) 16x 7y 64x A(,, B(, are two points. The locus of P which moves such that PA PB = 4 is y 3 0 ) y 3 0 y 3 0 4) y The perimeter of a triangle is 0 and the points, 3 and,3 The locus of the third vertex is x y 1 ) x y x y 1 4) none a 1 a The locus represented by x t, y t t t is x y a ) x y a x y a 4) x y a are two of the vertices of it.

3 16. The locus of the point a cos bsin,a sin b cos where 0 is x y a b ) x y 16xy x y a b 4) x y a b 17. If a point (x,y) = tan sin, tan sin, then the locus of (x,y) is / 3 /3 x y xy x y 4xy 4) x y 1xy x y 16xy 18. The locus of the point cos ec sin,sec cos where 0 is / 3 /3 x y xy 1 ) /3 / 3 x / a y / b 1 /3 /3 x y xy 1 /3 / 3 4) x / a y / b The locus of the point represented by x = 3 cos t sin t, y cos t sin t is x y x y 1 ) x y x y 1 4) The locus of the point represented by x t t 1, y t t 1 is x xy y x y 4 0 ) x xy y x y 4 0 x xy y x y 4 0 4) x xy y x y If a point P moves such that its distances from the point A(1, and the line x+y+ = 0 are equal then the locus of P is a straight line ) a pair of straight line a parabola 4) an ellipse. If p, x 1, x, x 3... and q, y 1, y, y 3... form two infinite AP s with common differences a and b x1 x... x y y... y respectively, then locus of P,, where and n n a x p b y q ) px a q y b px p a y q 4) bx p a y q 1 n

4 3. A straight rod of length 9 unit, slides with its ends A,B always on the x and y axes respectively. Then the locus of the centroid of OAB is x y 3 ) x y 9 x y 1 4) x y The ends of a rod of length / move on two mutually perpendicular lines. The locus of the point on the rod which divides it in the ratio 1: is 9x 34y ) 9x 34y 9x 36y 4 4) none of these 5. Locus of centroid of the triangle whose vertices are (a cos t, a sin t), (b sin t, b cos t) and (1,0), where t is a parameter, is 3x 1 3y a b ) 3x 1 3y a b 3x 1 3y a b 4) 3x 1 3y a b 6. A(a,0), B( a,0) are two points. If a point P moves such that PAB PBA = / the locus of P is x y a ) x y a 0 x xy y a 4) x y a 7. A(a,0), B( a,0) are two points. If a point P moves such that PAB PBA = then the locus of P is x xy tan y a ) x xy cot y a x xy tan y a 4) x xycot y a 8. Equation x y x y 4 represents A circle ) A pair of lines A parabola 4) An ellipse

5 LOCUS- SOLUTIONS 1. Ans.3 PA 5 x y 5 x y 5. Ans.1 Let P(x,y)be a point in the locus. Distance from x-axis = (distance from y-axis) 3. Ans.1 y x y 4x. A,3B,5. Let the point be Px, y 4. Ans.1 PA = PB PA PB x y 3 x y 5 4x 6y 13 4x 10y 9 x y Locus equation is (a 1 -a )x+(b 1 -b )y OA OB e OB OA 5. Ans.1 1 a b a b 1 1 Let P +(x,y), Given that PA :PB = 3:1 PA 3PB PA 9PB x 9 y 9 x 1 y 6. Ans.1 7. Ans.3 Let P(x,y) A(0,0) B(4, 8x 8y 7 x y 9 PA = 3PB 4PA =9PB 4x y 9 x 4 y 3 5x 5y 7x 54y 5 0 A(c,0) B(-c,0). Let the point be P(x,y) PA PB 4C x c y x c y 4c x y c x y c

6 8. Ans. A(, B(1,5)C(-1,) Let P(x,y) PA PB PC x y 3 x 1 y 5 x 1 y 4x 6y 13 x 10y 6 9. Ans. 1 x 4y 5 10x 8y 9 0 A(,B(-1, Let P=(x,y) APB 90 PA PB AB x y 3 x 1 y Ans Ans.3 x y x 4y 1 0 x y x 4y 1 0 Let P=(x,y), Given that area of PAB = 1sq. Unit x 3 y y x 4 6y x 4 x 3y 11 4 x 3 y x 3y 11 1 x 3y x 6xy 9y x 66y 3 0 let P=(x,y). Given that area of POA x 0 y 0 x 0 y

7 x y x y 6x 4y 36x 64y 9x 16y 1. Ans.1 9x 16y 0 The locus of P is x 4y Ans. 14. Ans.1 16x 7y 64x 48 0 The locus of P is 4x 1 16 x 4x 4 7y 11 4 y 3 1 y A, 3, B,3. Let the third vertex be P= (x, y). AB = Ans. 16. Ans.1 PA+PB+AB = 0 PA+PB=0-AB = 0-6 = 14 ( 4 x 4y The locus of P is a 1 1 x x t t t t a a 1 1 y y t t t t a 1 1 t t 4 t t 1 4y 4 x x y x y 4x 4y 4 4 x y a a a a a Let the point be (x,y) x, y a cos bsin,a sin bsin x a cos bsin, y a sin bcos

8 17. Ans.4 = a cos sin b sin cos x y a cos bsin a sin b cos x y a b x, y tan sin, tan sin 18. Ans.1 x tan sin, y tan sin xy tan sin tan sin tan sin tan 1 cos tan sin 4 tan.sin x y 16 tan.sin x y tan sin tan sin x y 16xy x, y cos ec sin,sec cos 19. Ans.3 0. Ans.1 x cos ec sin, y sec cos 1 sin cos x cos ec sin sin sin ; 1 cos sin y sec cos cos cos 4 cos sin 3 x y cos xy sin cos cos sin sin cos 4 3 sin /3 /3 x y xy cos sin x y cos t sin t, cos t sin t 3 / 3 /3 x y xy 1 x y x y x y t 1, x y t x y x xy y x y 4 0 x y 1 x y x xy y 4 1. Ans.3

9 . Ans.4 = x1 x... x n n n p a n 1 a n p a n 1 a a n 1 p 1 q b n 1 / 3. Ans. 1 p a b p a q q b Locus of P is bx p a y q Let a, b be the intercepts made by the rod of length Ans.3 P x y be the centroid of OAB. Then a = 3x 1, b = 3y 1. Let a b 81 9x 9y 81 x y 9. The locus of P is x +y = Let A(a,0) B(0,b) be the end points of a rod of length. 5. Ans. AB = a+b= Let P(h,k) divides AB in the ratio 1: P=(a/3,b/ h a / 3 a 3h / and k b / 3 b 3k 9h a b 9k 9h 4k 4 Locus of P is 9x 36y G = A+B+C (3x, 3y) = (a cost +bsint+1, a sin t-b cost) 3x a cos t bsin t 1,3y a sin t bcos t a cos t bsin t,3y a sin t bcos t

10 6. Ans.4 3x 1 3y a b y y tan, tan a x a x 7. Ans.4 cot cot 0 cotcot 1 0 y P(x,y) B(-a,0) O x Q A(a,0) x y a y y tan, tan a x a x tan tan tan tan 1 tan tan y y a x a x ya x a x y y a x y 1 a x a x y y P(x,y) y x a x a x 1 a x y y y B(-a,0) O xq A(a,0) x a x y xy cot x xy cot y a

11 8. Ans. Since x y x y 4 1 Assuming, x y x y 8x Dividing () by (, we have x y x y x 3 Adding ( & (, we have x y 4 x x y x

LOCUS. Definition: The set of all points (and only those points) which satisfy the given geometrical condition(s) (or properties) is called a locus.

LOCUS. Definition: The set of all points (and only those points) which satisfy the given geometrical condition(s) (or properties) is called a locus. LOCUS Definition: The set of all points (and only those points) which satisfy the given geometrical condition(s) (or properties) is called a locus. Eg. The set of points in a plane which are at a constant