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![In the xy co-ordinate system, a circle C is given by the equation [m]x](/forum/styles/gmatclub_light/imageset/icon_post_target_unread.svg "In the xy co-ordinate system, a circle C is given by the equation [m]x"){kind=icon}
18 Feb 2022, 06:31
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C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
65%
(hard)
Question Stats:
60% (02:24) correct
40%
(02:17)
wrong
based on 165
sessions
History
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In the xy co-ordinate system, a circle C is given by the equation \(x^2+y^2=36\) . A triangle PQR is inscribed in the circle C such that side PQ of the triangle PQR, lies on the x- axis. If vertex R of the triangle lies on the line \(y+x-6=0\). What is the area of the triangle PQR? (Equation of a circle is given by\( \left(x-a\right)^2+\left(y-b\right)^2=r^2\) , where a and b are x and y co-ordinates of center of the circle, respectively and r, is the radius of the circle.)
A. 18
B. \(18\sqrt{2}\)
C. 36
D. 72
E. 98
A. 18
B. \(18\sqrt{2}\)
C. 36
D. 72
E. 98
21 Feb 2022, 10:14
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Here is the OE
Solution
Step 1: Understand Question Statement
• \(x^2+y^2=36\) is circle C
• Triangle PQR is inscribed in the Circle C.
• Side PQ of the triangle PQR, lies on the x- axis.
• Vertex R of the triangle PQR lies on the line \(y+x-6=0\)
We need to find the area of triangle PQR.
\(Step 2: Define Methodology \)
• Center of C is (0,0) and radius is 6.
• We will find the intersection of the given line and Circle C to get vertex R.
• We will draw the diagram as per the information given in the question.
• Using the properties of triangle and circles, we will find the area PQR.
Step 3: Calculate the final answer
• Let’s find the intersection of C and the given line, substitute the value of y from the equation of the line into the equation of C:
• \(x^2+{(6-x)}^2=36\) ⟹ \(2x(x-6)=0\)
• Substituting the above values of x into equation of line, we get:
• ⟹y-0-6=0 ⟹ y=6 , Hence, the intersection point is (0,6)
• ⟹y-6-6=0 ⟹ y=0 , Hence the intersection point is (6,0)
• Based on above information we get below diagram:
• Since PQR is inscribed in the circle with one side PQ as its diagonal.
• OR, OP and OQ are radii of C.
• Since RO is perpendicular to PQ,
• Area of triangle PQR = \(\frac{1}{2}\ast\ base\ast\ height=\frac{1}{2}\ast\ PQ\ast\ RO=\frac{1}{2}\ast12\ast6=36 \)
Thus, the correct answer is Option C.
Solution
Step 1: Understand Question Statement
• \(x^2+y^2=36\) is circle C
• Triangle PQR is inscribed in the Circle C.
• Side PQ of the triangle PQR, lies on the x- axis.
• Vertex R of the triangle PQR lies on the line \(y+x-6=0\)
We need to find the area of triangle PQR.
\(Step 2: Define Methodology \)
• Center of C is (0,0) and radius is 6.
• We will find the intersection of the given line and Circle C to get vertex R.
• We will draw the diagram as per the information given in the question.
• Using the properties of triangle and circles, we will find the area PQR.
Step 3: Calculate the final answer
• Let’s find the intersection of C and the given line, substitute the value of y from the equation of the line into the equation of C:
• \(x^2+{(6-x)}^2=36\) ⟹ \(2x(x-6)=0\)
• Substituting the above values of x into equation of line, we get:
• ⟹y-0-6=0 ⟹ y=6 , Hence, the intersection point is (0,6)
• ⟹y-6-6=0 ⟹ y=0 , Hence the intersection point is (6,0)
• Based on above information we get below diagram:
• Since PQR is inscribed in the circle with one side PQ as its diagonal.
• OR, OP and OQ are radii of C.
• Since RO is perpendicular to PQ,
• Area of triangle PQR = \(\frac{1}{2}\ast\ base\ast\ height=\frac{1}{2}\ast\ PQ\ast\ RO=\frac{1}{2}\ast12\ast6=36 \)
Thus, the correct answer is Option C.
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14 Nov 2023, 10:32
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ktzsikka
How can we assume that the length of the base PQ is equal to the length of the diameter of the circle , it can be equal to diameter or maybe less.
So how did you think of taking it as equal to diameter?
