Let $A$, $B$, $C$ be 3 points on the parabola $y^2=4x$. Let $D$, $E$, $F$ be midpoints of $AB$, $BC$ and $AC$ respectively. If $G$ is the centroid of $\triangle DEF$ and normals to the parabola at $A$, $B$ and $C$ are concurrent at $H(5,1)$, then slope of the line $GH$ is
Let an ellipse have foci $S_1$ and $S_2$ with eccentricity $e = \frac{1}{2}$. For a point $P$ on the ellipse, let $\angle PS_1S_2 = \alpha$, $\angle PS_2S_1 = \beta$, $\angle S_1PS_2 = \gamma$. If $\cot\frac{\alpha}{2}$, $\cot\frac{\gamma}{2}$, $\cot\frac{\beta}{2}$ are in A.P., then $\cos(\alpha - \beta)$ is
Let $S=\{(x,y)\in\mathbb{R}\times\mathbb{R}: y^2\leq8x,\;y^2\geq32-8x,\;x+y-6\geq0,\;4x-3y-8\leq0,\;x\geq0,\;y\geq0\}$. If area $=A$, value of $21A+792$ is
Let an ellipse have foci $S_1$ and $S_2$ with eccentricity $e = \frac{1}{2}$. For a point $P$ on the ellipse, let $\angle PS_1S_2 = \alpha$, $\angle PS_2S_1 = \beta$, $\angle S_1PS_2 = \gamma$. If $\cot\frac{\alpha}{2}$, $\cot\frac{\gamma}{2}$, $\cot\frac{\beta}{2}$ are in A.P., then $\cos(\alpha - \beta)$ is
Number of correct statements is $k$. Then $2k$ is: I) Chord joining $P(at_1^2,2at_1)$ and $Q(at_2^2,2at_2)$ on $y^2=4ax$ passes through focus when $t_1t_2=-1$ and focal chord PQ $=|a|(t_1+1/t_1)^2\ge 4a$. II) $P=\{\theta:\sin\theta-\cos\theta=\sqrt{2}\cos\theta\}$ and $Q=\{\theta:\sin\theta+\cos\theta=\sqrt{2}\sin\theta\}$, then $P=Q$. III) If $a,b,c,d$ distinct nonzero reals: $(a^2+b^2+c^2)p^2-2(ab+bc+cd)p+(b^2+c^2+d^2)\le 0$, then $a,b,c,d$ are in GP.
Let $S=\{(x,y)\in\mathbb{R}\times\mathbb{R}: y^2\leq8x,\;y^2\geq32-8x,\;x+y-6\geq0,\;4x-3y-8\leq0,\;x\geq0,\;y\geq0\}$. If area $=A$, value of $21A+792$ is