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Question

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


  1. 4x + 5y - xy = 20

  2.  4x + 5y - xy = 21

  3. 4x + 5y + xy = 40

  4. 5x + 4y + xy = 61


Solution

The correct option is A

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=0

T is obtained by replacing x by x+x12Y by y+y12,x2by xx1,y2 by yy, and xy by xy1+yx12.

In our case, (x1,y1)=(5,4) andx225+y216=1 is the equation of curve

 S=x225+y2161 is the equation of curve

 S1=5225+42161=1

T=5x25+4y161=x5+y41

SS1=T2

 (x225+y2161)×1=(x5+y41)2

 x225+y2161=x225+y216+12x5y2+xy10

 2x5+y2xy10=2 or 4x+5yxy=20                

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