(D) Let the coordinates of point $A$ on the parabola $y^2=4x$ be $(t^2, 2t)$,where $a=1$.Since $O$ is the origin $(0,0)$,the midpoint $M(h, k)$ of $OA

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(D) Let the coordinates of point $A$ on the parabola $y^2=4x$ be $(t^2, 2t)$,where $a=1$.Since $O$ is the origin $(0,0)$,the midpoint $M(h, k)$ of $OA$ is given by:$h = \frac{0+t^2}{2} = \frac{t^2}{2} \implies t^2 = 2h$$k = \frac{0+2t}{2} = t \implies t = k$Substituting $t=k$ into $t^2=2h$,we get $k^2 = 2h$.Replacing $(h, k)$ with $(x, y)$,the locus is $y^2=2x$.

Class 11 Mathematics 10-2. Parabola, Ellipse, Hyperbola Parabola

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Let $O$ be the origin and $A$ be a point on the curve $y^2=4x$. Then the locus of the midpoint of $OA$ is:

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A
$x^2=4y$
B
$x^2=2y$
C
$y^2=16x$
D
$y^2=2x$

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10-2. Parabola, Ellipse, Hyperbola

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Parabola

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The tangents drawn from the ends of the latus rectum of the parabola ${y^2} = 12x$ meet at:

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What is the eccentricity of a parabola?

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Find the equation of the parabola with focus $(4, -3)$ and vertex $(4, -1)$.

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If $m$ is the slope of the normal to the parabola $y^2 + 4x + 4y = 0$ at the point $(-3, 2)$,which is one of the endpoints of a focal chord,find the value of $m$.

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Find the equation of the parabola that satisfies the following conditions: Vertex $(0, 0)$; focus $(3, 0)$.

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Let $O$ be the origin and $A$ be a point on the curve $y^2=4x$. Then the locus of the midpoint of $OA$ is:

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The tangents drawn from the ends of the latus rectum of the parabola ${y^2} = 12x$ meet at:

What is the eccentricity of a parabola?

Find the equation of the parabola with focus $(4, -3)$ and vertex $(4, -1)$.

If $m$ is the slope of the normal to the parabola $y^2 + 4x + 4y = 0$ at the point $(-3, 2)$,which is one of the endpoints of a focal chord,find the value of $m$.

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