Standard +0.3 This is a multi-step problem involving differentiation, normals, and circle equations, but each step follows standard AS-level procedures. Finding k requires equating the normal gradient to the given line's gradient (routine), then using the normal equation to find Q, and finally applying the circle equation formula. While it has multiple parts worth 10 marks, it requires no novel insight—just careful application of standard techniques in sequence, making it slightly easier than average.
A curve has equation \(y = kx^{\frac{1}{2}}\) where \(k\) is a constant.
The point \(P\) on the curve has \(x\)-coordinate 4.
The normal to the curve at \(P\) is parallel to the line \(2x + 3y = 0\) and meets the \(x\)-axis at the point \(Q\).
The line \(PQ\) is the radius of a circle centre \(P\).
Show that \(k = \frac{1}{2}\).
Find the equation of the circle. [10]
A curve has equation $y = kx^{\frac{1}{2}}$ where $k$ is a constant.
The point $P$ on the curve has $x$-coordinate 4.
The normal to the curve at $P$ is parallel to the line $2x + 3y = 0$ and meets the $x$-axis at the point $Q$.
The line $PQ$ is the radius of a circle centre $P$.
Show that $k = \frac{1}{2}$.
Find the equation of the circle. [10]
\hfill \mbox{\textit{OCR AS Pure 2017 Q8 [10]}}