The slope of the tangent must be the same as the of the curve; that is, .
The equation of the straight line is , and as it is satisfied for the values , , then ; also, its .
The and the of the point of contact must also satisfy both the equation of the tangent and the equation of the curve.
We have then
four equations in , , , .
Equations (i) and (ii) give .
Replacing and by their value in this, we get
,
which simplifies to , the solutions of which are: and . Replacing in (i), we get and respectively; the two points of contact are then , ; and , .
Note.—In all exercises dealing with curves, students will find it extremely instructive to verify the deductions obtained by actually plotting the curves.