    Question

# The point on parabola 2y=x2, which is nearest to the point (0,5) is

A
(4,8)
No worries! We‘ve got your back. Try BYJU‘S free classes today!
B
(1,1/2)
No worries! We‘ve got your back. Try BYJU‘S free classes today!
C
(22,4)
Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses
D
None of these
No worries! We‘ve got your back. Try BYJU‘S free classes today!
Open in App
Solution

## The correct option is C (2√2,4)Let (h,k) lie on the curve x2=2y which is nearest to the point (0,5)Since (h,k) lie on the curve x2=2y⇒(h,k) will satisfy the equation of curve x2=2y⇒ Putting x=h and y=k in equation we geth2=2k ......(1)We need to minimize the distance of a point (h,k) from (0,5)Let D be the distance between (h,k) and (0,5)D=√h2+(5−k)2From (1) we have h2=2kD=√2k+(5−k)2Differentiating w.r.t k we getdDdk=12√2k+(5−k)2×ddk(2k+(5−k)2)=12√2k+(5−k)2×[2+2(5−k)ddk(5−k)]=12√2k+(5−k)2×[2−2(5−k)]=−4+k√2k+(5−k)2Put dDdk=0⇒−4+k√2k+(5−k)2=0⇒−4+k=0∴k=4 is a point of minima.D is minimum when k=4We have h2=2k=2×4=8∴h=2√2Hence the required point is (h,k)=(2√2,4)  Suggest Corrections  0      Similar questions  Related Videos   Characteristics of Sound Waves
MATHEMATICS
Watch in App  Explore more