Let the arc AC of a circle subtend a right angle at the centre O. If the point B on the arc AC, divides the arc AC such that length of arcBC length of arcAB=51, and OC=αOA+βOB, then α+2(3−1)β is equal to
Held on 23 Jan 2025 · Verified 6 Jul 2026.
23
2−3
53
2+3
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Let ABC be an equilateral triangle with orthocenter at the origin and the side BC on the line $x+2 \sqrt{2} y=4$. If the co-ordinates of the vertex A are $(\alpha, \beta)$, then the greatest integer less than or equal to $|\alpha+\sqrt{2} \beta|$ is
In an equilateral triangle $PQR$, let the vertex $P$ be at $(3, 5)$ and the side $QR$ be along the line $x + y = 4$. If the orthocentre of the triangle $PQR$ is $(\alpha, \beta)$, then $9(\alpha + \beta)$ is equal to:
Let $\overrightarrow{\mathrm{c}}$ and $\overrightarrow{\mathrm{d}}$ be vectors such that $|\overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{d}}|=\sqrt{29}$ and $\overrightarrow{\mathrm{c}} \times(2 \hat{i}+3 \hat{j}+4 \hat{k})=(2 \hat{i}+3 \hat{j}+4 \hat{k}) \times \overrightarrow{\mathrm{d}}$. If $\lambda_{1}, \lambda_{2}\left(\lambda_{1}>\lambda_{2}\right)$ are the possible values of $(\vec{c}+\vec{d}) \cdot(-7 \hat{i}+2 \hat{j}+3 \hat{k})$, then the equation $\mathrm{K}^{2} x^{2}+\left(\mathrm{K}^{2}-5 \mathrm{~K}+\lambda_{1}\right) x y+\left(3 \mathrm{~K}+\frac{\lambda_{2}}{2}\right) y^{2}-8 x+12 y+\lambda_{2}=0$ represents a circle, for K equal to :
Consider the circle $C: x^2+y^2-6x-8y-11=0$. Let a variable chord AB of the circle C subtend a right angle at the origin. If the locus of the foot of the perpendicular drawn from the origin on the chord AB is the circle $x^2+y^2-\alpha x - \beta y - \gamma = 0$, then $\alpha + \beta + 2\gamma$ is equal to ________.
Consider the parabola $P : y^2 = 4kx$ and the ellipse $E : \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$. Let the line segment joining the points of intersection of $P$ and $E$, be their latus rectums. If the eccentricity of $E$ is $e$, then $e^2 + 2\sqrt{2}$ is equal to _____.
Work through every JEE Main Coordinate Geometry PYQ, year by year.
