A(2,3,5), B(alpha, 3,3) and C(7,5, beta) are | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Let $D$ be the midpoint of $BC$. The coordinates of $D$ are $\left(\frac{\alpha+7}{2}, \frac{3+5}{2}, \frac{3+\beta}{2}\right) = \left(\frac{\alph

Vedclass · Accepted Answer

-$9$

Vedclass · Answer

(A) Let $D$ be the midpoint of $BC$. The coordinates of $D$ are $\left(\frac{\alpha+7}{2}, \frac{3+5}{2}, \frac{3+\beta}{2}\right) = \left(\frac{\alpha+7}{2}, 4, \frac{3+\beta}{2}\right)$.The direction ratios of the median $AD$ are given by the difference of coordinates of $D$ and $A$:$\left(\frac{\alpha+7}{2} - 2, 4 - 3, \frac{3+\beta}{2} - 5\right) = \left(\frac{\alpha+3}{2}, 1, \frac{\beta-7}{2}\right)$.Since the median $AD$ is equally inclined to the coordinate axes,its direction cosines are equal,which implies that its direction ratios must be proportional to $(1, 1, 1)$.Thus,$\frac{\alpha+3}{2} = 1$ and $\frac{\beta-7}{2} = 1$.Solving for $\alpha$: $\alpha+3 = 2 \Rightarrow \alpha = -1$.Solving for $\beta$: $\beta-7 = 2 \Rightarrow \beta = 9$.Therefore,$\frac{\beta}{\alpha} = \frac{9}{-1} = -9$.

$A(2,3,5), B(\alpha, 3,3)$ and $C(7,5, \beta)$ are the vertices of a triangle. If the median through $A$ is equally inclined with the coordinate axes,then $\frac{\beta}{\alpha}=$

Similar Questions

If the origin is the centroid of the triangle $PQR$ with vertices $P(2a, 2, 6)$,$Q(-4, 3b, -10)$,and $R(8, 14, 2c)$,then the values of $a, b, c$ respectively are:

$ABCD$ is a parallelogram,$P$ is the mid-point of $AB$. If $R$ is the point of intersection of $AC$ and $DP$,then $R$ divides $AC$ internally in the ratio

If the points $A(1, 3, 5)$, $B(2, 4, 6)$, and $C(4, 5, k)$ form a right-angled triangle, then the number of possible values of $k$ is:

If $G(3, -5, r)$ is the centroid of $\triangle ABC$,where $A \equiv (7, -8, 1)$,$B \equiv (p, q, 5)$,and $C \equiv (q+1, 5p, 0)$ are vertices of the triangle $ABC$,then the values of $p, q, r$ are respectively:

Three vertices of a parallelogram $ABCD$ are $A(3, -1, 2)$,$B(1, 2, -4)$,and $C(-1, 1, 2)$. Find the coordinates of the fourth vertex $D$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module