Question

# Find the equation of pair of tangents to the ellipse x225+y216=1 from (5,4) 4x + 5y - xy = 20  4x + 5y - xy = 21 4x + 5y + xy = 40 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) and‘x225+y216=1 is the equation of curve ⇒ S=x225+y216−1 is the equation of curve ⇒ S1=5225+4216−1=1 T=5x25+4y16−1=x5+y4−1 SS1=T2  ⇒(x225+y216−1)×1=(x5+y4−1)2  ⇒x225+y216−1=x225+y216+1−2x5−y2+xy10  ⇒2x5+y2−xy10=2 or 4x+5y−xy=20

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