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Question

The line y = 2x + c touches the ellipse fraction numerator x to the power of 2 end exponent over denominator 16 end fraction plus fraction numerator y to the power of 2 end exponent over denominator 4 end fraction equals 1 if c is equal to

  1. 0    
  2. ± 2 square root of 17    
  3. c = ± square root of 15    
  4. c = ± square root of 17.    

hintHint:

If the line y = mx + c touches the ellipse x squared over a squared space plus space y squared over b squared space equals space 1, then c squared space equals space a squared m squared space plus space b squared.

The correct answer is: ± 2 square root of 17


    If, The line y = 2x + c touches the ellipse fraction numerator x to the power of 2 end exponent over denominator 16 end fraction plus fraction numerator y to the power of 2 end exponent over denominator 4 end fraction equals 1 if c is equal to c squared space equals space a squared m squared space plus space b squared
    fraction numerator x to the power of 2 end exponent over denominator 16 end fraction plus fraction numerator y to the power of 2 end exponent over denominator 4 end fraction equals 1 comparing with standard form
    a squared space equals space 16 comma space a space equals space 4
b squared space equals space 4 comma space b space equals space 2 space
m space equals space a minus b space equals space 2
    Substituting these values in c squared space equals space a squared m squared space plus space b squared, we get
    c squared space equals space a squared m squared space plus space b squared
    c squared space equals space 16 cross times 4 plus space 4
c squared space equals space 68
c space equals space plus-or-minus 2 square root of 17
    Thus, The line y = 2x + c touches the ellipse fraction numerator x to the power of 2 end exponent over denominator 16 end fraction plus fraction numerator y to the power of 2 end exponent over denominator 4 end fraction equals 1 if c is equal to plus-or-minus 2 square root of 17

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