Sergio Heath

2023-02-24

Find all points on the curve ${x}^{2}{y}^{2}+xy=2$ where the slope of the tangent line is -1.

a4dlity3v

Beginner2023-02-25Added 6 answers

Angle OBA = 90 degrees

(By theorem, the radius across the point of contact is perpendicular to the tangent at any point on a circle.)

Pythagoras theorem in right angled ABO

$\Rightarrow O{B}^{2}=O{A}^{2}-A{B}^{2}\phantom{\rule{0ex}{0ex}}{r}^{2}=(r+7.5{)}^{2}-{15}^{2}\phantom{\rule{0ex}{0ex}}r2={r}^{2}+56.25+15r-225\phantom{\rule{0ex}{0ex}}15r=168.75\phantom{\rule{0ex}{0ex}}r=11.25cm\phantom{\rule{0ex}{0ex}}\therefore Diameter=2r=2\times 11.25=22.5cm$

(By theorem, the radius across the point of contact is perpendicular to the tangent at any point on a circle.)

Pythagoras theorem in right angled ABO

$\Rightarrow O{B}^{2}=O{A}^{2}-A{B}^{2}\phantom{\rule{0ex}{0ex}}{r}^{2}=(r+7.5{)}^{2}-{15}^{2}\phantom{\rule{0ex}{0ex}}r2={r}^{2}+56.25+15r-225\phantom{\rule{0ex}{0ex}}15r=168.75\phantom{\rule{0ex}{0ex}}r=11.25cm\phantom{\rule{0ex}{0ex}}\therefore Diameter=2r=2\times 11.25=22.5cm$

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