The tangent at a point P on the curve xmyn=am+n meets the x-axis at A and y-axis at B. The triangle formed by AOB has centroid at Q, If O be the origin, then
A
AP:PB=n:m
Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses
B
AP:PB=m:n
No worries! We‘ve got your back. Try BYJU‘S free classes today!
C
locus of Q is (3xm)m(3yn)n=[a(m+n)]m+n
Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses
D
locus of Q is (3xm)n(3yn)m=[a(m+n)]m+n
No worries! We‘ve got your back. Try BYJU‘S free classes today!
Open in App
Solution
The correct options are AAP:PB=n:m C locus of Q is (3xm)m(3yn)n=[a(m+n)]m+n xmyn=am+n⇒mlnx+nlny=(m+n)lna Differentiating w.r.t. x mx+nyy′=0⇒y′=−mynx So, equation of the tangent at P(x,y) is Y−y=−mynx(X−x) Tangent meets the x-axis at A and y-axis at B. ∴A=((m+n)xm,0)B=(0,(m+n)yn)
Let P divides AB internally in the ratio of r:1 Then x=r⋅0+1⋅(m+n)xmr+1 ⇒(r+1)x=(1+nm)x⇒r=nm∴AP:PB=n:m
Let coordinates of Q is (h,k) ∴h=(m+n)x3m and k=(m+n)y3n ⇒(3hm)m⋅(3kn)n=(m+n)m+nxmyn⇒(3xm)m(3yn)n=[a(m+n)]m+n