PUC Karnataka Science Class 12Department of Pre-University Education, Karnataka
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Solution for The Point on the Curve Y2 = X Where Tangent Makes 45° Angle with X-axis is (A) (1/2, 1/4) (B) (1/4, 1/2) (C) (4, 2)(D) (1, 1) - PUC Karnataka Science Class 12 - Mathematics

Question

The point on the curve y2 = x where tangent makes 45° angle with x-axis is
(a) (1/2, 1/4)
(b) (1/4, 1/2)
(c) (4, 2)(d) (1, 1)

Solution

(b) (1/4, 1/2)
Let the required point be (x1, y1).
The tangent makes an angle of 45o with the x-axis.
∴ Slope of the tangent = tan 45o = 1

$\text { Since, the point lies on the curve } .$

$\text { Hence }, {y_1}^2 = x_1$

$\text { Now,} y^2 = x$

$\Rightarrow 2y\frac{dy}{dx} = 1$

$\Rightarrow \frac{dy}{dx} = \frac{1}{2y}$

$\text { Slope of the tangent } = \left( \frac{dy}{dx} \right)_\left( x_1 , y_1 \right) =\frac{1}{2 y_1}$

$\text { Given }:$

$\frac{1}{2 y_1} = 1$

$\Rightarrow 2 y_1 = 1$

$\Rightarrow y_1 = \frac{1}{2}$

$\text { Now,}$

$x_1 = {y_1}^2 = \left( \frac{1}{2} \right)^2 = \frac{1}{4}$

$\therefore \left( x_1 , y_1 \right) = \left( \frac{1}{4}, \frac{1}{2} \right)$

Is there an error in this question or solution?
Solution The Point on the Curve Y2 = X Where Tangent Makes 45° Angle with X-axis is (A) (1/2, 1/4) (B) (1/4, 1/2) (C) (4, 2)(D) (1, 1) Concept: Tangents and Normals.
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