Circle 1 Question 6
6. Let the orthocentre and centroid of a triangle be $A(-3,5)$ and $B(3,3)$, respectively. If $C$ is the circumcentre of this triangle, then the radius of the circle having line segment $A C$ as diameter, is
(a) $\sqrt{10}$
(b) $2 \sqrt{10}$
(c) $3 \sqrt{\frac{5}{2}}$
(d) $\frac{3 \sqrt{5}}{2}$
(2018 Main)
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Answer:
Correct Answer: 6. (c)
Solution:
- Key idea Orthocentre, centroid and circumcentre are collinear and centroid divide orthocentre and circumcentre in $2: 1$ ratio.
We have orthocentre and centroid of a triangle be $A(-3,5)$ and $B(3,3)$ respectively and $C$ circumcentre.
Clearly, $A B=\sqrt{(3+3)^{2}+(3-5)^{2}}=\sqrt{36+4}=2 \sqrt{10}$
We know that, $A B: B C=2: 1$
$\Rightarrow \quad B C=\sqrt{10}$
Now, $A C=A B+B C=2 \sqrt{10}+\sqrt{10}=3 \sqrt{10}$
Since, $A C$ is a diameter of circle.
$$ \begin{array}{lll} \therefore & r & =\frac{A C}{2} \\ \Rightarrow & r & =\frac{3 \sqrt{10}}{2}=3 \sqrt{\frac{5}{2}} \end{array} $$
