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# Solution for Find the Points on the Curve X2 + Y2 − 2x − 3 = 0 at Which the Tangents Are Parallel to the X-axis ? - CBSE (Science) Class 12 - Mathematics

ConceptMaximum and Minimum Values of a Function in a Closed Interval

#### Question

Find the points on the curve x2 + y2 − 2x − 3 = 0 at which the tangents are parallel to the x-axis ?

#### Solution

Let (x1, y1) be the required point.

$\text { Since the point lie on the curve } .$

$\text { Hence } {x_1}^2 + {y_1}^2 - 2 x_1 - 3 = 0 . . . \left( 1 \right)$

$\text { Now }, x^2 + y^2 - 2x - 3 = 0$

$\Rightarrow 2x + 2y \frac{dy}{dx} - 2 = 0$

$\therefore \frac{dy}{dx} = \frac{2 - 2x}{2y} = \frac{1 - x}{y}$

$\text { Now,}$

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

$\text { Slope of the tangent } = 0 (\text { Given )}$

$\therefore \frac{1 - x_1}{y_1} = 0$

$\Rightarrow 1 - x_1 = 0$

$\Rightarrow x_1 = 1$

$\text { From (1), we get }$

${x_1}^2 + {y_1}^2 - 2 x_1 - 3 = 0$

$\Rightarrow 1 + {y_1}^2 - 2 - 3 = 0$

$\Rightarrow {y_1}^2 - 4 = 0$

$\Rightarrow y_1 = \pm 2$

$\text { Hence, the points are }\left( 1, 2 \right)\text { and }\left( 1, - 2 \right).$

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#### APPEARS IN

Solution Find the Points on the Curve X2 + Y2 − 2x − 3 = 0 at Which the Tangents Are Parallel to the X-axis ? Concept: Maximum and Minimum Values of a Function in a Closed Interval.
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