The locus of all points on the curve y 2-4 a x - a sin x - a at which the tangent is parallel to x-axis isA- a parabolaB- an ellipseC- a straight lineD- a circle

The correct option is A a parabola∵ y^2 = 4a (x + a sin (x/a))....(i) ∴ 2y dy/dx = 4a (1 + cos (x/a)) Since, tangent is parallel to x axis ∴dy/dx = 0 ⇒ 1 + cos ...

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Standard XII

Mathematics

Global Maxima

The locus of ...

Question

The locus of all points on the curve $y^{2} = 4 a (x + a s i n (\frac{x}{a}))$ at which the tangent is parallel to x-axis is

a straight line

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a circle

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a parabola

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an ellipse

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Solution

The correct option is C a parabola
$∵ y^{2} = 4 a (x + a s i n (\frac{x}{a})) . . . . (i) ∴ 2 y \frac{d y}{d x} = 4 a (1 + c o s (\frac{x}{a}))$
Since, tangent is parallel to x-axis
$∴ \frac{d y}{d x} = 0 \Rightarrow 1 + c o s (\frac{x}{a}) = 0 \Rightarrow c o s (\frac{x}{a}) = - 1 c o s π ∴ \frac{x}{a} = 2 n π \pm π o r x = (2 n \pm 1) π a$
$∴$ From Eq. (i)

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The locus of all points on the curve y2=4a(x+a sin(xa)) at which the tangent is parallel to x-axis is

The correct option is C a parabola∵y2=4a(x+a sin(xa))....(i)∴2ydydx=4a(1+cos(xa)) Since, tangent is parallel to x-axis ∴dydx=0⇒1+cos(xa)=0⇒cos(xa)=−1cosπ∴xa=2nπ±π or x=(2n±1)πa ∴ From Eq. (i)

The locus of all points on the curve $y^{2} = 4 a (x + a s i n (\frac{x}{a}))$ at which the tangent is parallel to x-axis is