Write the angle between the curves y^2 = 4x and x^2 = 2y - 3 at t… - Vidyadip

Question

Write the angle between the curves $y^2 = 4x$ and $x^2 = 2y - 3$ at the point $(1, 2).$

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Answer

Given:
$y^2 = 4x ...(1)$
$x^2 = 2y − 3 ...(2)$
On differentiating (1) w.r.t.x, we get
$2\text{y}\frac{\text{dy}}{\text{dx}}=4$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\frac{2}{\text{y}}$
$\Rightarrow\text{m}_1=\Big(\frac{\text{dy}}{\text{dx}}\Big)_{(1,2)}=\frac{2}{2}=1$
On differentiating (2) w.r.t.x, we get
$2\text{x}=2\frac{\text{dy}}{\text{dx}}$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\text{x}$
$\Rightarrow\text{m}_2=\Big(\frac{\text{dy}}{\text{dx}}\Big)_{(1,2)}=1$
Thus, we get
$\tan\theta=\Big|\frac{\text{m}_1-\text{m}_2}{1+\text{m}_1\text{m}_2}\Big|$
$\Rightarrow\tan\theta=\Big|\frac{1-1}{1+1}\Big|$
$\Rightarrow\tan\theta=0$
$\Rightarrow\theta=0$

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