Chapter 5: Problem 34
Slope of secant \(=\frac{9 a^{2}-a^{2}}{3 a-a}=4 a\) \(\frac{d y}{d x}=2 x=4 a\) \(x=2 a\) \(y=4 a^{2}\)
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\(x y^{2}=1\) \(y^{2}+2 x y \frac{d y}{d x}=0\) \(\frac{d y}{d x}=\frac{-y}{2 x}\) \(-\frac{d x}{d y}=\frac{2 x}{y}=\frac{2}{y^{3}}\) \(y-y_{1}=\frac{2}{y_{1}^{3}}\left(x-x_{1}\right)\) \(+y_{1}^{4}=2 x_{1}\) \(y_{1}^{6}=2\) \(y_{1}=\pm 2^{1 / 6}\) \(x_{1}=\pm 2^{-\sqrt{3}}\)
\(y=\left(\frac{x}{2}-a\right)^{2}+a-2\) \(4(y-(a-2))=(x-2 a)^{2}\) Vertex \(\Rightarrow \mathrm{h}=2 \mathrm{a}, \mathrm{k}=\mathrm{a}-2\) Locus of vertex \(\Rightarrow \mathrm{y}=\frac{\mathrm{x}}{2}-2\) \(\Rightarrow 2 y=x-4\)
eqn of normal $\sin t / 2 y-a\left[2 \sin \frac{t}{2} \sin t-\sin 2 t \sin \frac{t}{2}\right]$ $$ =x \cos \frac{t}{2}-a\left[2 \cos t \cos \frac{t}{2}+\cos 2 t \operatorname{tos} \frac{t}{2}\right] $$ $\Rightarrow x \cos \frac{t}{2}-y \sin \frac{t}{2}=a\left[\begin{array}{c}2 \cos t \cos \frac{t}{2}-2 \sin t \sin \frac{t}{2} \\ +\cos 2 \operatorname{t} \cos \frac{t}{2}+\sin 2 t \sin \frac{t}{2}\end{array}\right]$ \(=a\left[2 \cos \frac{3 t}{2}+\cos \frac{3 t}{2}\right]\) \(=3 a \cos \frac{3 t}{2}\)
Let \(\mathrm{P}\) be \(\left(\mathrm{t}^{2}, 2 \mathrm{t}\right)\) \(y^{2}=4 x\) \(2 y \frac{d y}{d x}=4\) \(\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{1}{\mathrm{t}}\) If \(Q\) is a pt where normal meets the parabola again wit \(\mathrm{t}_{1}\) as parameter \(\mathrm{t}_{1}=-\mathrm{t}-\frac{2}{\mathrm{t}}\) Coordinate of $Q \rightarrow\left(\left(t+\frac{2}{t}\right)^{2},-2 t-\frac{4}{t}\right)$ Distance \(=\sqrt{\left(\mathrm{t}+\frac{2}{\mathrm{t}}\right)^{4}+4\left(\mathrm{t}+\frac{2}{\mathrm{t}}\right)^{2}}\) Distance $=\left(\mathrm{t}+\frac{2}{\mathrm{t}}\right) \sqrt{\left(\mathrm{t}+\frac{2}{\mathrm{t}}\right)^{2}+4}$ Min Distance \(=2 \sqrt{2} \sqrt{12}=4 \sqrt{6}\)
slope of normal \(\Rightarrow 3 x-y+3=0\) $$ x=0 \& y=3 $$ Pt of normal \(=(0,3)\) \(\frac{d y}{d x}=\frac{-1}{3}=f^{\prime}(0)\) $\lim _{x \rightarrow 0} \frac{x^{2}}{f\left(x^{2}\right)+4 f\left(7 x^{2}\right)-5 f\left(4 x^{2}\right)}$ $\lim _{x \rightarrow 0} \frac{2 x}{x\left[2 f^{\prime}\left(x^{2}\right)+56 f^{\prime}\left(7 x^{2}\right)-40 f^{\prime}\left(4 x^{2}\right)\right]}$ \(=\frac{2}{-6}=\frac{-1}{3}\)
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