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