| Directions: Each of these questions contains two statements: Assertion [A] and Reason [R]. Each of these questions also has four alternative choices, any one of which is the correct answer. You have to select one of the codes [a], [b], [c] and [d] given below. |
| Assertion If the points A (4, 3) and B (x, 5) are on the circle with centre O (2, 3), then find the value of x is 2. |
| Reason If three points (0, 0), \[\left( 3,\,\sqrt{3} \right)\]and \[\left( 3,\,\lambda \right)\]form an equilateral triangle, then \[\lambda \] equals to \[\pm \sqrt{2}\]. |
A) A is true, R is true; R is a correct explanation for A.
B) A is true, R is true; R is not a correct explanation for A.
C) A is true; R is False.
D) A is false; R is true.
Correct Answer: C
Solution :
| Since, A and B lie on the circle having centre O. |
| \[OA=OB\] |
| \[\Rightarrow \,\,\sqrt{{{\left( 4-2 \right)}^{2}}+{{\left( 3-3 \right)}^{2}}}\] |
| \[=\sqrt{{{\left( x-2 \right)}^{2}}+{{\left( 5-3 \right)}^{2}}}\] |
| \[\Rightarrow \,2=\sqrt{{{\left( x-2 \right)}^{2}}+4}\] |
| \[\Rightarrow \,\,\,4={{\left( x-2 \right)}^{2}}+4\] |
| \[\Rightarrow \,\,\,{{\left( x-2 \right)}^{2}}=0\Rightarrow \,x=2\] |
| Let the given points are A (0, 0), \[B\left( 3,\,\sqrt{3} \right)\]and \[C\left( 3,\,\lambda \right)\] |
| Since, \[\Delta ABC\]is an equilateral triangle, therefore \[AB=AC\] |
| \[\Rightarrow \,\,\,\sqrt{{{\left( 3-0 \right)}^{2}}+{{\left( \sqrt{3}-0 \right)}^{2}}}\] |
| \[=\sqrt{{{\left( 3-0 \right)}^{2}}+{{\left( \lambda -0 \right)}^{2}}}\] |
| \[\Rightarrow \,\,\,9+3=9+{{\lambda }^{2}}\,\Rightarrow {{\lambda }^{2}}=3\] |
| \[\Rightarrow \,\,\,\lambda =\pm \sqrt{3}\] |
| Assertion is true but Reason is false. |
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