CAT 2023 Question Paper | Quant Slot 2

CAT Questions

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

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Video Explanation

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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 \( = \frac { a b } { 2 } \times \frac { 4 r ^ { 2 } } { \left( a ^ { 2 } + b ^ { 2 } \right) } = \frac { 2 a b r ^ { 2 } } { a ^ { 2 } + b ^ { 2 } } \)

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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.