CAT 2023 Question Paper | Quant Slot 2

CAT 2023 Quant was dominated by Algebra followed by Arithmetic. In Arithmetic, the questions were dominated by topics like Speed-time-distance, Mixture and Alligations. This year, there was a surprise. The questions from Geometry were relatively on the lower side as compared to the previous years. There were 8 TITA Qs this year. Overall this section was at a medium level of difficulty.

Question 18 : A triangle is drawn with its vertices on the circle $C$ such that one of its sides is a diameter of $C$ and the other two sides have their lengths in the ratio $a: b$. If the radius of the circle is $r$, then the area of the triangle is

Video Explanation

Explanatory Answer

Since the angle subtended by the diameter on the circle is a right-angle, such a triangle inscribed in a circle with the diameter as one of its sides will be right angled.

“…and the other two sides have their lengths in the ratio a: b.”
Let the two sides be ax and bx.
The area of the triangle = $\frac { 1 } { 2 } \times a x \times b x = \frac { a b } { 2 } x ^ { 2 }$
By the Pythagoras theorem, $( a x ) ^ { 2 } + ( b x ) ^ { 2 } = ( 2 r ) ^ { 2 }$
$\left( a ^ { 2 } + b ^ { 2 } \right) \times x ^ { 2 } = 4 r ^ { 2 }$
$x ^ { 2 } = \frac { 4 r ^ { 2 } } { \left( a ^ { 2 } + b ^ { 2 } \right) }$
∴The area of the triangle $= \frac { a b } { 2 } \times \frac { 4 r ^ { 2 } } { \left( a ^ { 2 } + b ^ { 2 } \right) } = \frac { 2 a b r ^ { 2 } } { a ^ { 2 } + b ^ { 2 } }$

The question is " A triangle is drawn with its vertices on the circle $C$ such that one of its sides is a diameter of $C$ and the other two sides have their lengths in the ratio $a: b$. If the radius of the circle is $r$, then the area of the triangle is "

The answer is '$\frac{2 a b r^2}{a^2+b^2}$'

Choice D is the correct answer.

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