Maths-
General
Easy

Question

The number of values of c such that the straight line y=4x+c touches the curve left parenthesis x to the power of 2 end exponent divided by 4 right parenthesis plus y to the power of 2 end exponent equals 1 is‐

  1. 0    
  2. 1    
  3. 2    
  4. infinite    

The correct answer is: 2


    We need to find the number of values of c such that the straight line y=4x+c touches the curve left parenthesis x to the power of 2 end exponent divided by 4 right parenthesis plus y to the power of 2 end exponent equals 1.
    S i n c e space t h e space l i n e space y equals 4 x plus c space t o u c h e s space t h e space e l l i p s e fraction numerator space x squared over denominator 4 end fraction plus y squared equals 1 comma space w e space s u b s t i t u t e space
y space a s space 4 x plus c space i n t o space t h e space e l l i p s e.

W e space g e t space fraction numerator space x squared over denominator 4 end fraction plus 16 space x squared plus space 8 c x plus c squared equals 1
space x squared plus 64 space x squared plus 32 c x plus 4 c squared minus 4 equals 0
T h e space d i s c r i m i n a n t space o f space t h i s space q u a d r a t i c space s h o u l d space b e space z e r o space sin c e space w e space h a v e space o n l y space o n e space v a l u e space o f space x.

open parentheses 32 c close parentheses squared equals 4 cross times 65 cross times left parenthesis 4 c squared minus 4 right parenthesis
1024 c squared equals 16 cross times 65 cross times left parenthesis c squared minus 1 right parenthesis
64 c squared equals 65 cross times left parenthesis c squared minus 1 right parenthesis
64 c squared equals 65 c squared minus 65
c squared equals 65
T h u s comma space w e space h a v e space t w o space v a l u e s space o f space c.

    Therefore, there are two values of c.

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