36 + Interview Questions in LOCUS IN GENERAL AWARENESS Page 1 InterviewSolution

1.	If P (x,y) and Q (1,4) are the points on the circle whose centre is :C(5,7) and radius is 5 cm, then find the locus of P.
Answer» Correct Answer - `x^(2) + y^(2) - 10x - 14 y + 49 = 0`

Discussion

2.	If two lines intersect at P at right angular and pass through A(1,1) and B(1,0) respectively when what is the locus of P?
Answer» Correct Answer - `x^(2) + y^(2) - 2x - y + 1 =0`

Discussion

3.	In `Delta PAB` D,E and F are the mid-points of PA, AB and BP respectively. The area of DEF is 8 sq.units. If A is (2,5) and B is (3,4), then what is the locus of P?
Answer» Correct Answer - `x + y - 71 0; x + y + 57 = 0`

Discussion

4.	The locus of point equidistant from three fixed points is a single point. The three points are________A. collinearB. non-collinearC. coincidentalD. None of these
Answer» Correct Answer - B Three points form a triangle

Discussion

5.	If P is a point on the circle with AB as a diameter, where A and B are (0, 2) and (2,4) respectively, then he locus of P is ____
Answer» Correct Answer - `x^(2) + y^(2) = 2x - 6y + 8 = 0`

Discussion

6.	The locus of a point which is collinear with the two given points is ________A. a circleB. a triangleC. a straight lineD. a parabola
Answer» Correct Answer - C

Discussion

7.	The locus of a point moving in a space which is at a constant distance from a fixed point in space is called a ____A. squareB. sphereC. circleD. triangle
Answer» Correct Answer - B Recall the definition.

Discussion

8.	The locus of the centre of a circle the touches the given circle externally is a _______A. curveB. straight lineC. circleD. helix
Answer» Correct Answer - C The required locus is a circle

Discussion

9.	The solid formed when a right triangle is rotated about one of the sids containing the right angle in a _____A. prismB. cylinderC. coneD. sphere
Answer» Correct Answer - C When a right triangle is rotated about one of its perpendicular sides, the other perpendicular side acts as radius of the base and the hypotenuse acts as the slant height of solid. (ii) The top of a solid is a point (vertex).

Discussion

10.	The locus of a points, which is at a distance of 8 units from (0, - 7), is ______A. `x^(2) + y^(2) + 6x + 14y - 15 = 0`B. `x^(2) + y^(2) = 14 g - 15 = 0`C. `y^(2) + 14 g - 8 = 0`D. `x^(2) + y^(2) + 14x + 14 y - 15 = 0`
Answer» Correct Answer - B

Discussion

11.	In `Delta ABC, /_ A = /_ B + /_ C`, then the circum centre is at ______A. AB. BC. CD. the mid points of BC
Answer» Correct Answer - D Recall the propterties of right triangle with respect to geometric centres.

Discussion

12.	If A and B are two fixed point, then the loucs of a point P, such that `PA^(2) + PB^(2) = AB^(2)` is a/an ______A. circle with AB as the diameterB. right triangle with `/_ P = 90^(@)`C. semi with AB as the diameterD. circle with AB as the diameter, excluding points A and B
Answer» Correct Answer - D Angle in semicircle is `90^(@)`

Discussion

13.	The loucs of the point which is equidistant from the three determined by the sides of a triangles is _______A. the in-centreB. the ex-centreC. the ortho-centreD. either (a) or (b)
Answer» Correct Answer - D Recall the definitions of geometric centres.

Discussion

14.	The locus of a rectangle, when the rectangle is rotated about one of its sides is a ______A. plineB. sphereC. coneD. cylinder
Answer» Correct Answer - D The required triangle locus is a cylinder.

Discussion

15.	The locus of a point equidistant from the points does not exists. This implies that the three points are _____A. collinearB. non-collinearC. coincidentalD. None of these
Answer» Correct Answer - A

Discussion

16.	The locus of the point in a plane which is equidistant from two intersecting lines is _____
Answer» Correct Answer - angle bisectors of the two pairs of vertically opposite angle of intersecting liens.

Discussion

17.	The locus of the point equidistant (in a plane) from the three vertices of a triangle is ____
Answer» Correct Answer - circum-centre

Discussion

18.	Centroid of triangle is :(A) Intersection point of ⊥ bisectors of sides of triangle.(B) Intersection point of angle bisectors of triangle.(C) Intersection point of medians of triangle.(D) Intersection point of altitudes of triangle.
Answer» Answer is (C) Intersection point of medians of triangle.

Discussion

19.	The locus of a point, which is equidistant from (2,6) and (2, 8) is ______A. `y = 7`B. `x = 7`C. `x = 2`D. `y = 2`
Answer» Correct Answer - A Let A = (2,6) B = (2,8) The required locus is the perpendicular bisector of `overline(AB)` Let K be the point of intensection of `overline(AB)` and its perpendicular bisector. `K = ((2 + 2)/(2), (8 + 6)/(2)) = (2,7)` Slope of `overline(AB) = ((8 - 6)/(2 -2))`is not defined. `:. overline(AB)` is parallel to Y-axis. The required line is parallel to X-axis `:.` The required line is Y = 7 (From Eqs. (1) and (2)).

Discussion

20.	In given figure, PS, bisector of ∠P intersects side QR at point S. SN ⊥ PQ and SM ⊥ PQ are drawn. Prove that SN = SM.
Answer» Given : In ∆PQS, PS is bisector of ∠P and SN ⊥ PQ and SM ⊥ PR. To Prove : SN = SM Proof : In ∆PSN and ∆PSM PS = PS (Common) ∠PNS = ∠PMS (Each 90° because SN ⊥ PQ and SM ⊥ PR) ∠NPS = ∠MPS (Since PS, bisector of ∠P) by A.A.S. congruence rule ∆ASP ≅ ∆PSM ∴ SN = SM Corresponding sides of congruent angles are equal.

20.

In given figure, PS, bisector of ∠P intersects side QR at point S. SN ⊥ PQ and SM ⊥ PQ are drawn. Prove that SN = SM.

Answer»

Given : In ∆PQS, PS is bisector of ∠P and SN ⊥ PQ and SM ⊥ PR.

To Prove : SN = SM

Proof : In ∆PSN and ∆PSM

PS = PS (Common)

∠PNS = ∠PMS (Each 90° because SN ⊥ PQ and SM ⊥ PR)

∠NPS = ∠MPS (Since PS, bisector of ∠P)

by A.A.S. congruence rule

∆ASP ≅ ∆PSM

∴ SN = SM

Corresponding sides of congruent angles are equal.

Discussion

21.	The locus of a point in space, which is always at a distance of 4cm from a fixed point.
Answer» The locus of a point in space is the surface of the sphere whose centre is the fixed point and radius equal to 4 cm.

Discussion

22.	Find the locus of the point which is equidistant from sides AB and AD of a rhombus ABCD.
Answer» Correct Answer - The requrired locus is diagonal AC.

Discussion

23.	Construct a triangle ABC, with AB = 7cm, BC = 8cm and ∠ABC = 60°. Locate by construction the point P such that:(i) P is equidistant from B and C.(ii) P is equidistant from AB and BC.Measure and record the length of PB.
Answer» Steps of Construction: 1) Draw a line segment AB = 7 cm. 2) Draw angle ∠ABC = 60° with the help of compass. 3) Cut off BC = 8 cm. 4) Join A and C. 5) The triangle ABC so formed is the required triangle. i) Draw the perpendicular bisector of BC. The point situated on this line will be equidistant from B and C. ii) Draw the angle bisector of ∠ABC . Any point situated on this angular bisector is equidistant from lines AB and BC. The point which fulfills the condition required in (i) and (ii) is the intersection point of bisector of line BC and angular bisector of ∠ABC. P is the required point which is equidistant from AB and AC as well as from B and C. On measuring the length of line segment PB, it is equal to 4.5 cm.

23.

Construct a triangle ABC, with AB = 7cm, BC = 8cm and ∠ABC = 60°. Locate by construction the point P such that:(i) P is equidistant from B and C.(ii) P is equidistant from AB and BC.Measure and record the length of PB.

Answer»

Steps of Construction:

1) Draw a line segment AB = 7 cm.

2) Draw angle ∠ABC = 60° with the help of compass.

3) Cut off BC = 8 cm.

4) Join A and C.

5) The triangle ABC so formed is the required triangle.

i) Draw the perpendicular bisector of BC. The point situated on this line will be equidistant from B and C.

ii) Draw the angle bisector of ∠ABC . Any point situated on this angular bisector is equidistant from lines AB and BC.

The point which fulfills the condition required in (i) and (ii) is the intersection point of bisector of line BC and angular bisector of ∠ABC.

P is the required point which is equidistant from AB and AC as well as from B and C.

On measuring the length of line segment PB, it is equal to 4.5 cm.

Discussion

24.	The locus of point.s located at equal distance from two fixed points is called :(A) line segment(B) circle(C) ⊥ bisector of line segment(D) two points
Answer» Answer is (C) ⊥ bisector of line segment

Discussion

25.	In given figure D, E and F are midpoint of sides BC, CA and AB respectively. If AB = 4.3 cm, BC = 5.6 cm and AC = 3.9 cm then find perimeter of thefollowing:(i) ∆DEF and(ii) quadrilateral BDEF.
Answer» We know that line segment joining the mid-points of two sides is half the third side. Thus in ∆DEF, DE = 1/2AB EF = 1/2BC DF = 1/2AC ∴ Perimeter of ∆DEF = DE + EF + DF = 1/2AB + 1/2BC + 1/2AC = 1/2(AB + BC + AC) = 1/2(4.3 + 5.6 + 3.9) = 1/2(13 . 8) = 6.9 cm (ii) Perimeter of quadrilateral BDEF = BD + DE + EF + 8F = 1/2BC + 1/2AB + 1/2BC + 1/2AB = AB + BC = 4.3 + 5.6 = 9.9 cm

25.

In given figure D, E and F are midpoint of sides BC, CA and AB respectively. If AB = 4.3 cm, BC = 5.6 cm and AC = 3.9 cm then find perimeter of thefollowing:(i) ∆DEF and(ii) quadrilateral BDEF.

Answer»

We know that line segment joining the mid-points of two sides is half the third side.

Thus in ∆DEF,

DE = 1/2AB

EF = 1/2BC

DF = 1/2AC

∴ Perimeter of ∆DEF = DE + EF + DF

= 1/2AB + 1/2BC + 1/2AC

= 1/2(AB + BC + AC)

= 1/2(4.3 + 5.6 + 3.9)

= 1/2(13 . 8)

= 6.9 cm

(ii) Perimeter of quadrilateral BDEF = BD + DE + EF + 8F

= 1/2BC + 1/2AB + 1/2BC + 1/2AB

= AB + BC

= 4.3 + 5.6

= 9.9 cm

Discussion

26.	In given figure, at common base BC, two isosceles triangles ∆PBC and ∆QBC lie on both sides of BC. Prove that line joining P and Q bisects line BC at 90°.
Answer» Given : ∆PBC and ∆QBC are two isosceles triangles which lie on both sides of base BC. Here, PB = PC and BQ = CQ To Prove : ∠POB = ∠POC = 90° or ∠QOB = ∠QOC = 90° Proof : In ∆PBC, PB = PC (Given) ∴ ∠PBO = ∠PCO (Equal sides) PO = PO (Common) by S.A.S. congruence of ∆POB ≅ ∆POC (Corresponding sides of corresponding triangles are equal) ⇒ ∠PBO = ∠POC ….(i) We Know that ∠PBO + ∠POC = 180° ∠PBO + ∠POB = 180° [From equation (i)] 2∠POB = 180° ∠POB = 180°/2 = 90° ∠PBO = ∠POC = 90° Similarly, ∠QOB = ∠QOC = 90° Thus, PQ, bisects BC at 90°

26.

In given figure, at common base BC, two isosceles triangles ∆PBC and ∆QBC lie on both sides of BC. Prove that line joining P and Q bisects line BC at 90°.

Answer»

Given : ∆PBC and ∆QBC are two isosceles triangles which lie on both sides of base BC.

Here, PB = PC and BQ = CQ

To Prove : ∠POB = ∠POC = 90°

or ∠QOB = ∠QOC = 90°

Proof : In ∆PBC,

PB = PC (Given)

∴ ∠PBO = ∠PCO (Equal sides)

PO = PO (Common)

by S.A.S. congruence of

∆POB ≅ ∆POC (Corresponding sides of corresponding triangles are equal)

⇒ ∠PBO = ∠POC ….(i)

We Know that

∠PBO + ∠POC = 180°

∠PBO + ∠POB = 180° [From equation (i)]

2∠POB = 180°

∠POB = 180°/2 = 90°

∠PBO = ∠POC = 90°

Similarly, ∠QOB = ∠QOC = 90°

Thus, PQ, bisects BC at 90°

Discussion

27.	What will be the locus of mid-points of equal chords of a circle?(A) Circle(B) Chord(C) Distance of center from chord(D) Center point
Answer» Answer is (A) Circle

Discussion

28.	In the following criterian, in which criterion two triangles will not be congruent?(A) All corresponding sides are equal(B) All corresponding angles are equal(C) Two corresponding sides and angle between them are equal.(D) All corresponding angles and one corresponding side are equal.
Answer» Answer is (B) All corresponding angles are equal

Discussion

29.	The centroid of triangle divides each median in the ratio :(A) 2 : 1(B) 1 : 2(C) 2 : 3(D) 3 : 2
Answer» Answer is (A) 2 : 1

Discussion

30.	Triangle whose othocentre is vertices of triangle is called :(A) Right angled triangle(B) Equilateral triangle(C) Isosceles triangle(D) None of the above
Answer» Answer is (A) Right angled triangle

Discussion

31.	Distance of circumcentre from vertices of triangle will be :(A) equal(B) different(C) zero(D) different, equal
Answer» Answer is (A) equal

Discussion

32.	Intersection point of three altitudes of a triangles is called:(A) Circumcentre(B) incentre(C) Orthocentre(D) Centroid
Answer» Answer is (C) Orthocentre

Discussion

33.	Triangle whose orthocentre, circumcentre and in centre are concurrent is called :(A) Equilateral triangle(B) Right angled triangle(C) Isosceles triangle(D) None of the above
Answer» Answer is (A) Equilateral triangle

Discussion

34.	Any point on the perpendicular bisector of a line segment joining two point is ________ from the two points.
Answer» Correct Answer - equidistant

Discussion

35.	Centroid divides the median from the vertex in the ratio_____
Answer» Correct Answer - `2 : 1`

Discussion

36.	In figure, BF = FC, ∠BAE – ∠CAE and ∠ADE – ∠GEC = 90°. Then name one altitude, the median one angle bisector and one perpendicular bisector of side of triangle.
Answer» Altitude = AD Median = AF Angle bisector = AE perpendicular bisector = GE.

36.

In figure, BF = FC, ∠BAE – ∠CAE and ∠ADE – ∠GEC = 90°. Then name one altitude, the median one angle bisector and one perpendicular bisector of side of triangle.

Answer»

Altitude = AD

Median = AF

Angle bisector = AE

perpendicular bisector = GE.

Discussion

Explore topic-wise InterviewSolutions in .

If P (x,y) and Q (1,4) are the points on the circle whose centre is :C(5,7) and radius is 5 cm, then find the locus of P.

If two lines intersect at P at right angular and pass through A(1,1) and B(1,0) respectively when what is the locus of P?

In `Delta PAB` D,E and F are the mid-points of PA, AB and BP respectively. The area of DEF is 8 sq.units. If A is (2,5) and B is (3,4), then what is the locus of P?

The locus of point equidistant from three fixed points is a single point. The three points are________A. collinearB. non-collinearC. coincidentalD. None of these

If P is a point on the circle with AB as a diameter, where A and B are (0, 2) and (2,4) respectively, then he locus of P is ____

The locus of a point which is collinear with the two given points is ________A. a circleB. a triangleC. a straight lineD. a parabola

The locus of a point moving in a space which is at a constant distance from a fixed point in space is called a ____A. squareB. sphereC. circleD. triangle

The locus of the centre of a circle the touches the given circle externally is a _______A. curveB. straight lineC. circleD. helix

The solid formed when a right triangle is rotated about one of the sids containing the right angle in a _____A. prismB. cylinderC. coneD. sphere

The locus of a points, which is at a distance of 8 units from (0, - 7), is ______A. `x^(2) + y^(2) + 6x + 14y - 15 = 0`B. `x^(2) + y^(2) = 14 g - 15 = 0`C. `y^(2) + 14 g - 8 = 0`D. `x^(2) + y^(2) + 14x + 14 y - 15 = 0`

In `Delta ABC, /_ A = /_ B + /_ C`, then the circum centre is at ______A. AB. BC. CD. the mid points of BC

If A and B are two fixed point, then the loucs of a point P, such that `PA^(2) + PB^(2) = AB^(2)` is a/an ______A. circle with AB as the diameterB. right triangle with `/_ P = 90^(@)`C. semi with AB as the diameterD. circle with AB as the diameter, excluding points A and B

The loucs of the point which is equidistant from the three determined by the sides of a triangles is _______A. the in-centreB. the ex-centreC. the ortho-centreD. either (a) or (b)

The locus of a rectangle, when the rectangle is rotated about one of its sides is a ______A. plineB. sphereC. coneD. cylinder

The locus of a point equidistant from the points does not exists. This implies that the three points are _____A. collinearB. non-collinearC. coincidentalD. None of these

The locus of the point in a plane which is equidistant from two intersecting lines is _____

The locus of the point equidistant (in a plane) from the three vertices of a triangle is ____

Centroid of triangle is :(A) Intersection point of ⊥ bisectors of sides of triangle.(B) Intersection point of angle bisectors of triangle.(C) Intersection point of medians of triangle.(D) Intersection point of altitudes of triangle.

The locus of a point, which is equidistant from (2,6) and (2, 8) is ______A. `y = 7`B. `x = 7`C. `x = 2`D. `y = 2`

In given figure, PS, bisector of ∠P intersects side QR at point S. SN ⊥ PQ and SM ⊥ PQ are drawn. Prove that SN = SM.

The locus of a point in space, which is always at a distance of 4cm from a fixed point.

Find the locus of the point which is equidistant from sides AB and AD of a rhombus ABCD.

Construct a triangle ABC, with AB = 7cm, BC = 8cm and ∠ABC = 60°. Locate by construction the point P such that:(i) P is equidistant from B and C.(ii) P is equidistant from AB and BC.Measure and record the length of PB.

The locus of point.s located at equal distance from two fixed points is called :(A) line segment(B) circle(C) ⊥ bisector of line segment(D) two points

In given figure D, E and F are midpoint of sides BC, CA and AB respectively. If AB = 4.3 cm, BC = 5.6 cm and AC = 3.9 cm then find perimeter of thefollowing:(i) ∆DEF and(ii) quadrilateral BDEF.

In given figure, at common base BC, two isosceles triangles ∆PBC and ∆QBC lie on both sides of BC. Prove that line joining P and Q bisects line BC at 90°.

What will be the locus of mid-points of equal chords of a circle?(A) Circle(B) Chord(C) Distance of center from chord(D) Center point

In the following criterian, in which criterion two triangles will not be congruent?(A) All corresponding sides are equal(B) All corresponding angles are equal(C) Two corresponding sides and angle between them are equal.(D) All corresponding angles and one corresponding side are equal.

The centroid of triangle divides each median in the ratio :(A) 2 : 1(B) 1 : 2(C) 2 : 3(D) 3 : 2

Triangle whose othocentre is vertices of triangle is called :(A) Right angled triangle(B) Equilateral triangle(C) Isosceles triangle(D) None of the above

Distance of circumcentre from vertices of triangle will be :(A) equal(B) different(C) zero(D) different, equal

Intersection point of three altitudes of a triangles is called:(A) Circumcentre(B) incentre(C) Orthocentre(D) Centroid

Triangle whose orthocentre, circumcentre and in centre are concurrent is called :(A) Equilateral triangle(B) Right angled triangle(C) Isosceles triangle(D) None of the above

Any point on the perpendicular bisector of a line segment joining two point is ________ from the two points.

Centroid divides the median from the vertex in the ratio_____

In figure, BF = FC, ∠BAE – ∠CAE and ∠ADE – ∠GEC = 90°. Then name one altitude, the median one angle bisector and one perpendicular bisector of side of triangle.