The curve described parametrically by y -2 at and x - at 2- where - is a non-zero constant- representsA- an ellipseB- a parabolaC- a hyperbolaD- a circle

The correct option is B a parabolax=at^2, y=2at Eliminating the parameter t, t=y2α ⇒ x=a(y2a)^2 ⇒ y^2=4ax ⇒ y^2 4a x=0 Comparing with ax^2+2hxy+by^2+2gx+2fy + ...

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Standard XII

Mathematics

Distinguishing between Conics from General Equation and Eccentricity

The curve des...

Question

The curve described parametrically by $y = 2 a t$ and $x = a t^{2},$ where $α$ is a non-zero constant, represents

an ellipse

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a parabola

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a hyperbola

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a circle

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Solution

The correct option is B a parabola
$x = a t^{2}, y = 2 a t$
Eliminating the parameter $t,$
$t = \frac{y}{2 α} \Rightarrow x = a {(\frac{y}{2 a})}^{2} \Rightarrow y^{2} = 4 a x \Rightarrow y^{2} - 4 a x = 0$

Comparing with $a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0$
$a = 0, b = 1, c = f = h = 0, g = - 2 a$
$∴ △ = ∣ ∣ ∣ ∣ \begin{matrix} 0 & 0 & - 2 a 0 & 1 & 0 - 2 a & 0 & 0 \end{matrix} ∣ ∣ ∣ ∣ = - 4 a^{2} \neq 0$

Also, $h^{2} - a b = 0$
Hence, given curve represents a parabola.

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The curve described parametrically by y=2at and x=at2, where α is a non-zero constant, represents

The correct option is B a parabolax=at2,y=2at Eliminating the parameter t, t=y2α⇒x=a(y2a)2⇒y2=4ax⇒y2−4ax=0 Comparing with ax2+2hxy+by2+2gx+2fy+c=0 a=0,b=1,c=f=h=0,g=−2a ∴△=∣∣ ∣∣00−2a010−2a00∣∣ ∣∣=−4a2≠0 Also, h2−ab=0 Hence, given curve represents a parabola.

The curve described parametrically by $y = 2 a t$ and $x = a t^{2},$ where $α$ is a non-zero constant, represents