Maths-
General
Easy

Question

Three normals are drawn to the parabola y to the power of 2 end exponent equals 4 x from the point (c, 0) These normals are real and distinct when

  1. c=0    
  2. c=1    
  3. c=2    
  4. c=3    

hintHint:

A normal in geometry is a piece of geometry that is perpendicular to another piece of geometry, such as a line, ray, or vector. For instance, the (infinite) line perpendicular to the tangent line to the curve at the point is the normal line to a plane curve at the point. Here we have to find these normals are real and distinct at what c.

The correct answer is: c=3


    At a specific point on a curve, tangents and normals behave just like any other straight lines. The normal runs perpendicular to the curve, and a tangent is parallel to the curve at the point. Like any other straight line, the equation of tangent and normal can be evaluated.
    Here we have given parabola of y2 = 4x.
    N o w space w e space k n o w space t h a t space a n y space n o r m a l space o f space t h e space p a r a b o l a space i s space g i v e n space b y space t h e space f o r m u l a colon
y equals m x minus 2 a m minus a m cubed
I n space t h i s space c a s e space a equals 1.
N o w space t h a t space w e space h a v e space g i v e n space t h a t space i t space p a s s e s space t h r o u g h space t h e space p o i n t space left parenthesis c comma 0 right parenthesis comma space s o space w e space g e t colon
m left parenthesis m squared plus 2 minus c right parenthesis equals 0
N o w space i n space o r d e r space t o space g e t space n o r m a l space a s space r e a l space a n d space d i s t i n c t comma space w e space h a v e colon
2 minus c less than 0
c greater than 2
S o space t h e space v a l u e space o f space c space i s space 3.

    So here we used the concept of tangents and normals, we found the slope and then went to proceed with the final answer. The normal to a parabola is perpendicular to the tangent of the parabola. So here the value of c is 3.

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