Tag: basic geometrical concepts and shapes

Quizzes (14) Questions (3)

Questions Related to basic geometrical concepts and shapes

Multiple choice maths basic geometrical concepts and shapes construction of line segment and circle of given radius construction related to lines constructing line segment

Four distinct points $(2K , 3K), (1 , 0), (0 , 1)$ and $(0 , 0)$ lie on a circle when

  1. all values of $K$ are integral
  2. $0 < K < 1$
  3. $K < 0$
  4. For one values of $K$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let A = $(0,0)$ , B$(0,1)$ , C$(2k,3k)$ , D$(1,0)$

As we can see

$AD \perp AB$

$\angle A = 90$

$\angle C = 90$

$\implies m _{BC} \times m _{DC} = -1$

$\dfrac{3k - 1}{2k} \times {3k}{2k - 1} = -1$

$\implies k(9k - 3) = -(4k - 2)k$

$k = 0 , 9k + 4k = 2+ 3 \implies 13k = 5 \implies k = \dfrac{5}{13}$

But if $k = 0$, C will be $(0,0)$ which is A

$k = \dfrac{5}{13} $ only one point.

Multiple choice maths basic geometrical concepts and shapes construction of line segment and circle of given radius construction related to lines constructing line segment

A ray of light passing through the point A$(1, 2, 3)$, strikes the plane $x + y + z = 12$ at B on reflection passes through point C$(3, 5, 9)$. The coordinates of point B are 

  1. $(2, 5, 5)$
  2. $(-4, 6, 10)$
  3. $(-7, 0, 19)$
  4. $(0, -5, 17)$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
Let image of point $A(1,2,3)$ about $x+y+z=12$ be $D(p,q,r)$ then,
$\cfrac{p-1}{1}=\cfrac{q-2}{1}=\cfrac{r-3}{1}=-2\cfrac{1+2+3-12}{1^2+1^2+1^2} \\ D(p,q,r)=D(5,6,7) $

Line joining $CD$
$\cfrac{x-5}{2}=\cfrac{y-6}{1}=\cfrac{z-7}{-2}=\lambda$
Coordinate of $B(2\lambda+5,\lambda+6,-2\lambda+7)$
This lies on plane $2\lambda+5+\lambda+6-2\lambda+7=12 \Rightarrow \lambda=-6$
$B(2\lambda+5,\lambda+6,-2\lambda+7)=B(-7,0,19)$