Question
Download Solution PDFThe point on the curve y2 = x, where the tangent makes an angle of \(\rm \frac{\pi}{4}\) with the x-axis, is:
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFConcept:
The angle made by the tangent to the curve y = f(x) at a point (a, b), with the x-axis, is given by m = tan θ = \(\rm \left[\frac{dy}{dx}\right]_{(a, b)}\).
Calculation:
The given curve is y2 = x.
⇒ \(\rm 2y\frac{dy}{dx}=1\)
⇒ \(\rm \frac{dy}{dx}=\frac{1}{2y}\)
For the tangent to make an angle of \(\rm \frac{\pi}{4}\), we must have:
tan \(\rm \frac{\pi}{4}\) = \(\rm \frac{dy}{dx}=\frac{1}{2y}\)
⇒ 1 = \(\rm \frac{1}{2y}\)
⇒ y = \(\rm \frac{1}{2}\)
Also, x = y2 = \(\rm \frac{1}{4}\).
The required point is \(\left( \frac 1 4, \frac 1 2 \right)\).
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