Home
Class 12
MATHS
ABC is an equilateral triangle where AB ...

ABC is an equilateral triangle where AB = a and P is any point in its plane such that PA = PB + PC. Then `(PA^(2)+PB^(2)+PC^(2))/(a^(2))` is

A

3

B

`(sqrt(3))/(4)`

C

`(3)/(4)`

D

2

Text Solution

Verified by Experts

The correct Answer is:
D
PA = PB + PC and ABC is equiliateral
`therefore` P lies on arc BC of circumcircle of `Delta ABC`

`therefore angle BPC = 120^(@)`
Using cosine rule in `Delta BPC`
`cos 120^(@)=(PB^(2)+PC^(2)-BC^(2))/(2PB.PC)`
`rArr -PB.PC=PB^(2)+PC^(2)-a^(2)`
`rArr 2PB.PC=2a^(2)-2(PB^(2)+PC^(2))`
Again, PA = PB + PC
`rArr PA^(2)=PB^(2)+PC^(2)+2PB.PC`
`rArr PA^(2)=PB^(2)+PC^(2)+2a^(2)-2(PB^(2)+PC^(2))`
`(PA^(2)+PB^(2)+PC^(2))/(a^(2))=2`
Promotional Banner

Topper's Solved these Questions

  • SOLUTIONS AND PROPERTIES OF TRIANGLE

    CENGAGE|Exercise Multiple Correct Answers Type|13 Videos

  • SOLUTIONS AND PROPERTIES OF TRIANGLE

    CENGAGE|Exercise Comprehension Type|6 Videos

  • SETS AND RELATIONS

    CENGAGE|Exercise Question Bank|15 Videos

  • STATISTICS

    CENGAGE|Exercise JEE Previous Year|10 Videos

Similar Questions

Explore conceptually related problems

Let ABCD be a rectangle and P be any point in its plane.Show that AP^(2)+PC^(2)=PB^(2)+PD^(2)

If P be any point in the plane of square ABCD, prove that PA^(2)+PC^(2)=PB^(2)+PD^(2)

IF P is any point within a Delta ABC, then PA+CP is

If G be the centroid of a triangle ABC and P be any other point in the plane prove that PA^(2)+PB^(2)+PC^(2)=GA^(2)+GB^(2)+GC^(2)+3GP^(2)

There is a point P in a rectangle ABCD, such that PA = 4, PD = 5, PB = 8, find PC.

An equilateral triangle ABC is inscribed in a circle of radius r if P be any point on the circle then find the value of PA^(2)+PB^(2)+PC^(2)

[" An equilateral triangle "ABC" is inscribed in a circle.P is any point on the minor are BC.Prove "],[" that "PA=PB+PC]

ABC is a right triangle with right angle at B, if P is a mid point of AC show that PB=PA=PC=(1)/(2)(AC)

In a triangle ABC with angleA=90^(@) , P is a point on BC such that PA : PB = 3 : 4 If AB=sqrt7" and " AC=sqrt5 then BP : PC is-