First slide

Tangents and normals

Question

The point on the curve $y^{2} = x$ where tangent makes an angle of $45 °$ with the X-axis is

Moderate

A

$(\frac{1}{2}, \frac{1}{4})$
B

$(\frac{1}{4}, \frac{1}{2})$
C

$(4, 2)$
D

$(2, - 2)$

Solution

Let $(α, β)$ be the required point $β^{2} = α$ and ${(\frac{d y}{d x})}_{(α, β)} = \tan 45 °$
$(∵ y^{2} = x, ∴ 2 y \frac{d y}{d x} = 1)$

$\Rightarrow \frac{1}{2 β} = 1 \Rightarrow β = \frac{1}{2}$

Also, $α = β^{2} = {(\frac{1}{2})}^{2} = \frac{1}{4}$

Hence, the required point is $(\frac{1}{4}, \frac{1}{2})$ .

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