Question

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

• A. $x = \frac { 1 } { 4 }$
• B. $y = \frac { 1 } { 4 }$
• C. $y = \frac { 1 } { 2 }$
• D. $y = 1$

Answer 1

Answer: (C)

$y = \frac { 1 } { 2 }$

Answer 2

Let the equation of line parallel to x-axis be

$y = \lambda$ .....(i)

Solving (i) with the cuve $y = \sqrt { x }$ .....(ii)

We get $P \left( \lambda ^ { 2 } , \lambda \right)$ the point of intersection at P

$\therefore$ Slope of (ii) is, m=

$\therefore$(i) and (ii) intersect at P, at $45 ^ { \circ }$

$\therefore$ $\tan ^ { - 1 } \left( \frac { m - 0 } { 1 + m .0 } \right) = \pm 45 ^ { \circ }$ .

⇒ $m = \left( \frac { 1 } { 2 \lambda } \right) = \pm 1$ ⇒ $\lambda = \pm \frac { 1 } { 2 }$

$\therefore$ The equation of line is $y = \frac { 1 } { 2 }$ or $y = \frac { - 1 } { 2 }$ but $y = \frac { - 1 } { 2 }$ is not given, hence the required line is $y = \frac { 1 } { 2 }$.