JEE Main · 2021 · 31 Aug 2021 · Shift 1 · Parabola

Let O be the vertex of the parabola $x^2=4y$ and Q be any point on it. Let the locus of the point P , which divides the line segment OQ internally in the ratio 2: 3 be the conic C . Then the equation of the chord of C, which is bisected at the point $(1,2)$, is:

Let $y^2=12x$ be the parabola with its vertex at O. Let P be a point on the parabola and A be a point on the x-axis such that $\angle OPA=90^{\circ }$. Then the locus of the centroid of such triangles OPA is :

Let one end of a focal chord of the parabola $y^2=16x$ be (16, 16). If $P(\alpha ,\beta )$ divides this focal chord internally in the ratio $5:2$, then the minimum value of $\alpha +\beta$ is equal to :

If the chord joining the points ${\mathrm{P}}_1\left({\mathrm{x}}_1,{\mathrm{y}}_1\right)$ and ${\mathrm{P}}_2\left({\mathrm{x}}_2,{\mathrm{y}}_2\right)$ on the parabola $y^2=12x$ subtends a right angle at the vertex of the parabola, then $x_1{x}_2-y_1{y}_2$ is equal t …

Let the locus of the mid-point of the chord through the origin O of the parabola ${\mathrm{y}}^2=4\mathrm{x}$ be the curve S . Let P be any point on S . Then the locus of the point, which internally divides OP in the ratio $3:1$, is :

An equilateral triangle OAB is inscribed in the parabola ${\mathrm{y}}^2=4\mathrm{x}$ with the vertex O at the vertex of the parabola. Then the minimum distance of the circle having AB as a diameter from the origin is