1
You visited us 1 times! Enjoying our articles? Unlock Full Access!
Question

# P is a point inside the triangle ABC from which the length of perpendiculars drawn of the sides of lengths a,b,c are respectively p,q and r. Determine the position of p if ap+bq+cr is minimum.

A
circumcentre of the triangle
No worries! We‘ve got your back. Try BYJU‘S free classes today!
B
incentre of the triangle
Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses
C
centroid of the triangle
No worries! We‘ve got your back. Try BYJU‘S free classes today!
D
orthocentre of the triangle
No worries! We‘ve got your back. Try BYJU‘S free classes today!
Open in App
Solution

## The correct option is B incentre of the triangleΔ=ΔPBC+ΔPCA+ΔPAB =12(ap+bq+cr)=k =constant as Δ is given (Δ means area of triangle)Now z=ap+bq+cr will be minimum if y=k(ap+bq+cr) =12(ap+bq+cr)(ap+bq+cr) is minimum.⇒y=12[a2+b2+c2+ab(pq+qp)+bc(qr+rq)+ca(rp+pr)] Now we know that A.M.>G.M.pq+qp≥2(pq×qp)1/2=2 Equality when pq+qp=0⇒p2=q2⇒p=q ∴y≥12[a2+b2+c2+2ab+2bc+2ca] Hence the minimum value of y is 12(a+b+c)2 when p=q=r.In this case point P will be at the incentre of ΔABC.

Suggest Corrections
0
Join BYJU'S Learning Program
Related Videos
MATHEMATICS
Watch in App
Explore more
Join BYJU'S Learning Program