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Question

Find the equation of pair of tangents to the ellipse x225+y216=1 from (5, 4).


  1. 5x + 4y + xy = 61

  2. 4x + 5y - xy = 21               

  3. 4x + 5y - xy = 20 

  4. 4x + 5y + xy = 40


Solution

The correct option is C

4x + 5y - xy = 20 


The equation of a pair of tangent to any second degree curve
S=ax2+2hxy+by2+29x+2fy+c=0
is given by SS1=T2, where
S1=ax21+2hx1y1+by21+29x1+2fy1+c
T is obtained by replacing x by x+x12y by y+y12,x2 by xx1,y2 by yy, and xy by xy1+yx12.
In our case, (x1,y1)=(5,4) and x225+y216=1 is the equation of curve
S=x225+y2161 is the equation of curve
S1=5225+42161=1T=5x25+4y161=x5+y41SS1=T2(x225+y2161)×1=(x5+y41)2x225+y2161=x225+y216+12x5y2+xy102x5+y2xy10=2 or 4x+5yxy=20


 

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