E-TN03 · Three normals, including a vertical one
A normal to \(y=x^2\) at \((a,a^2)\) passes through \(P=(0,5)\).
For \(a\ne0\), show that the normal is \(\displaystyle y=-\frac{x}{2a}+a^2+\frac12\).
Find every normal through P. Include the case excluded in part (a).
Find the area of the triangle formed by the two oblique normals and the x-axis.
Complete-answer checking, worked solutions and indicative written criteria.
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