First slide

Equation of line in Straight lines

Question

The line which is parallel to $x$ -axis and crosses the curve $y = \sqrt{x}$ at an angle of $45^{\circ}$ , is

Moderate

A

$x = 1 / 4$
B

$y = 1 / 4$
C

$y = 1 / 2$
D

$y = 1$

Solution

The equation of a line parallel to $x$ -axis is $y = k .$
Solving this equation with $y = \sqrt{x},$ we get $x = k^{2} .$
Thus, the line $y = k$ intersect with $y = \sqrt{x}$ at $(k^{2}, k) .$
It is given that the line $y = k$ intersects the curve $y = \sqrt{x}$ at an angle
or $n / 4 .$ This means that the slope of the tangent to $y = \sqrt{x}$ at
$({k|}^{2}, k)$ is $\tan (\pm π / 4) = \pm 1 .$
$\Rightarrow {(\frac{d y}{d x})}_{(k^{2}, k)} = \pm 1 \Rightarrow {(\frac{1}{2 \sqrt{x}})}_{(k^{2}, k)} = \pm 1 \Rightarrow k = \pm \frac{1}{2}$
Hence, the required lines are $y = \pm \frac{1}{2}$

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