Let $A(-2,-1)$, $B(1,0)$, $C(\alpha,\beta)$ and $D(\gamma,\delta)$ be the vertices of a parallelogram $ABCD$. If the point $C$ lies on $2x-y=5$ and the point $D$ lies on $3x-2y=6$, then the value of $|\alpha+\beta+\gamma+\delta|$ is equal to
If $(\alpha,\beta)$ is the orthocentre of $\triangle ABC$ with $A(3,-7)$, $B(-1,2)$, $C(4,5)$, then $9\alpha-6\beta+60$ is equal to
Let A(6, 8), B(10 cos \alpha, -10 sin \alpha) and C(-10 sin \alpha, 10 cos \alpha), be the vertices of a triangle. If L(a, 9) and G(h, k) be its orthocenter and centroid respectively, then (5a - 3h + 6k + 100 sin 2\alpha) is equal to ______ -.
Line $\frac{x}{a} + \frac{y}{b} = 1$ cuts the coordinate axes at $A(a, 0)$ and $B(0, b)$ and the line $\frac{x}{a'} + \frac{y}{b'} = -1$ at $A'(-a', 0)$ and $B'(0, -b')$. If the points $A$, $B$, $A'$, $B'$ are concyclic then the orthocentre of the triangle $ABA'$ is:
Let $\left(5,\dfrac{a}{4}\right)$ be the circumcenter of a triangle with vertices $A(a,-2)$, $B(a,6)$ and $C\left(\dfrac{a}{4},-2\right)$. Let $\alpha$ denote the circumradius, $\beta$ denote the area and $\gamma$ denote the perimeter of the triangle. Then $\alpha+\beta+\gamma$ is
Let n Cr-1 = 28, n Cr = 56 and n Cr+1 = 70 . Let A(4 cos t, 4 sin t), B(2 sin t, -2 cos t) and C (3r - n, r - n - 1) 2 be the vertices of a triangle ABC , where t is a parameter. If (3x - 1) + (3y) = \alpha, is the locus of the centroid of 2 2 triangle ABC , then \alpha equals
Let $A(0,1)$, $B(1,1)$ and $C(1,0)$ be the midpoints of sides of a triangle with incentre $D$. If the focus of $y^2=4ax$ through $D$ is $(\alpha+\beta\sqrt{2},0)$, then $\dfrac{\alpha}{\beta^2}$ is equal to
Let ${}^nC_{r-1} = 28$, ${}^nC_r = 56$ and ${}^nC_{r+1} = 70$. Let $A(4\cos t, 4\sin t)$, $B(2\sin t,-2\cos t)$ and $C(3r-n, r^2-n-1)$ be the vertices of a triangle $ABC$, where $t$ is a parameter. If $(3x-1)^2+(3y)^2 = \alpha$ is the locus of the centroid of triangle $ABC$, then $\alpha$ equals
Let the diagonals of a convex quadrilateral ABCD intersect at point P, and let a, b, c, d denote the lengths of sides AB, BC, CD, and DA respectively. Then: Diagonals of quadrilateral ABCD are perpendicular if and only if \(a^2 + c^2 = b^2 + d^2\).
Vertices of a parallelogram ABCD are \(A(3, 1)\), \(B(13, 6)\), \(C(13, 21)\) and \(D(3, 16)\). If a line passing through the origin divides the parallelogram into two congruent parts then the slope of the line is:
Let $A(6,8)$, $B(10\cos\alpha,-10\sin\alpha)$ and $C(-10\sin\alpha,10\cos\alpha)$ be the vertices of a triangle. If $L(a,9)$ and $G(h,k)$ be its orthocenter and centroid respectively, then $(5a-3h+6k+100\sin 2\alpha)$ is equal to ____.
Let \(x_1 = x_1,\ x_2 = x_1 r,\ x_3 = x_1 r^2\) and \(y_1 = y_1,\ y_2 = y_1 r,\ y_3 = y_1 r^2\). Which of the following is true about the points \(A \equiv (x_1, y_1)\), \(B \equiv (x_2, y_2)\), \(C \equiv (x_3, y_3)\)?