MY CART (5)

Use Coupon: CART20 and get 20% off on all online Study Material

ITEM
DETAILS
MRP
DISCOUNT
FINAL PRICE
Total Price: Rs.

⇐ BackProceed to Checkout ⇒
There are no items in this cart.
Continue Shopping
Menu
Grade: 12 
        
What is the least distance between (0,12) and (a(t^2),2at) where a is constant and t is variable.
3 months ago

Answers : (1)

Sukant Kumar
27 Points 
							
Let the distance between the given points be D, then we know that
 
D=\sqrt{(at^2)^2 + (2at-12)^2}
 
D=\sqrt{(a^2t^4) + (4a^2t^2+144-48at)}
 
D=\sqrt{a^2t^4+ 4a^2t^2-48at+144}
 
Let, as when f(t) will be minimum and then D will also be minimum,
 
f(t)=a^2t^4+4a^2t^2-48at
 
f`(t)=4a^2t^3+8a^2t-48a=a(4at^3+8at-48)
 
f``(t)=a(12at^2+8a) > 0, \forall, t \in R,
 
hence f’(t) is monotonically increasing. Minimum of f(t) is when f`(t) reaches zero. The answer thus depends on the value of a and cannot be resolved easily in terms of a. So let us say that at t1 the value of f`(t) is zero and f(t) is minimum, right? Then,
 
f`(t_1)=4at^3+8at-48=0 \rightarrow f(t) = a^2t_1^4+4a^2t_1^2-48at_1=at_1(at_1^3)+4a^2t_1^2-48at_1=at_1(\frac{48-8at_1}{4})+4a^2t_1^2-48at_1=2a^2t_1^2-36at_1=2at_1(at_1-18))
Hence, the expression doesn’t resolve to a constant value and the minimum distance will not have a uniform expression for different values of a.
 
3 months ago
Think You Can Provide A Better Answer ?
Answer & Earn Cool Goodies

Other Related Questions on Analytical Geometry

Find the locus of the focus of an ellipse which touches the x axis and whose length of the major and... Answer & Earn Cool Goodies
Please solve the problem in the attachment... .. ........ . .................. ..... . ... ...... ..... Answer & Earn Cool Goodies
find equation of circle whose center lies on positive direction of y axis at distance 6 from origin...
I didnt understand the equation for shifted parabola. Plz help
How can I prove whether a circle is real or imaginary when only radius was given?
View all Questions »


Course Features

  • 731 Video Lectures
  • Revision Notes
  • Previous Year Papers
  • Mind Map
  • Study Planner
  • NCERT Solutions
  • Discussion Forum
  • Test paper with Video Solution


Course Features

  • 53 Video Lectures
  • Revision Notes
  • Test paper with Video Solution
  • Mind Map
  • Study Planner
  • NCERT Solutions
  • Discussion Forum
  • Previous Year Exam Questions


Ask Experts

Have any Question? Ask Experts

Post Question

 
 
Answer ‘n’ Earn
Attractive Gift
Vouchers To Win!!! Click Here for details