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06 Aug 2022, 20:59
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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 \((x-a)^2 + (y-b)^2 = r^2\) , 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√2
C. 36
D. 72
E. 98
A. 18
B. 18√2
C. 36
D. 72
E. 98
06 Aug 2022, 21:20
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Official Explanation
Step 1: Understand Question Statement
1. \(x^2 + y^2\)\(\) = 36 is circle C
2. Triangle PQR is inscribed in the Circle C.
3. Side PQ of the triangle PQR, lies on the x- axis.
4. 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
1. Center of C is (0,0) and radius is 6.
2. We will find the intersection of the given line and Circle C to get vertex R.
3. We will draw the diagram as per the information given in the question.
4. 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⟹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)
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 = \(\frac{1}{2}\) * base * height = \(\frac{1}{2}\) * PQ * RQ = \(\frac{1}{2}\) * 12 * 6 = 36
Thus, the correct answer is Option C.
Step 1: Understand Question Statement
1. \(x^2 + y^2\)\(\) = 36 is circle C
2. Triangle PQR is inscribed in the Circle C.
3. Side PQ of the triangle PQR, lies on the x- axis.
4. 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
1. Center of C is (0,0) and radius is 6.
2. We will find the intersection of the given line and Circle C to get vertex R.
3. We will draw the diagram as per the information given in the question.
4. 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⟹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)
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 = \(\frac{1}{2}\) * base * height = \(\frac{1}{2}\) * PQ * RQ = \(\frac{1}{2}\) * 12 * 6 = 36
Thus, the correct answer is Option C.
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