The equation of a curve is $$\(y=2+\sqrt{25-x^{2}}\)$$. Find the coordinates of the point on the curve at which the gradient is $$\(\frac{4}{3}\)$$. ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................
Exam No：9709_w20_qp_11 Year：2020 Question No：6
Answer:
\(\frac{\mathrm{d} y}{\mathrm{~d} x}=\left[\frac{1}{2}\left(25-x^{2}\right)^{-1 / 2}\right] \times[-2 x]\)
\(\frac{-x}{\left(25-x^{2}\right)^{1 / 2}}=\frac{4}{3} \rightarrow \frac{x^{2}}{25-x^{2}}=\frac{16}{9}\)
\(16\left(25-x^{2}\right)=9 x^{2} \rightarrow 25 x^{2}=400 \rightarrow x=(\pm) 4\)
When \(x=-4, y=5 \rightarrow(-4,5)\)
\(\frac{-x}{\left(25-x^{2}\right)^{1 / 2}}=\frac{4}{3} \rightarrow \frac{x^{2}}{25-x^{2}}=\frac{16}{9}\)
\(16\left(25-x^{2}\right)=9 x^{2} \rightarrow 25 x^{2}=400 \rightarrow x=(\pm) 4\)
When \(x=-4, y=5 \rightarrow(-4,5)\)
Knowledge points:
1.1.3 solve quadratic equations, and quadratic inequalities, in one unknown (By factorising, completing the square and using the formula.)
1.7.1 understand the gradient of a curve at a point as the limit of the gradients of a suitable sequence of chords, and use the notations for first and second derivatives (Only an informal understanding of the idea of a limit is expected.)
1.7.2 use the derivative of (for any rational ), together with constant multiples, sums and differences of functions, and of composite functions using the chain rule
Solution:
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