Question:
The parabola $y = -x^2 + 16$ intersects the $x$-axis at points A and B. Further, the parabola intersects the line $y = 7$ at points M and N. Then the area of the quadrilateral with vertices A, B, M and N is
The parabola $y = -x^2 + 16$ intersects the $x$-axis at points A and B. Further, the parabola intersects the line $y = 7$ at points M and N. Then the area of the quadrilateral with vertices A, B, M and N is
Show Hint
Plotting or visualizing the points on a Cartesian plane quickly reveals that the quadrilateral is a symmetric trapezoid.
This allows you to bypass complex polygon area formulas and use the simple trapezoid area formula directly.
This allows you to bypass complex polygon area formulas and use the simple trapezoid area formula directly.
Updated On: Jun 16, 2026
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The Correct Option is A
Solution and Explanation
Step 1: Understanding the Question:
The question asks for the area of the quadrilateral formed by the intersection points of a parabola with the $x$-axis and a horizontal line.
By determining the coordinates of these four points, we can identify the shape of the quadrilateral and calculate its area.
Step 2: Key Formula or Approach:
The intersection points on the $x$-axis are found by setting $y = 0$.
The intersection points with the line $y = 7$ are found by setting $y = 7$.
The area of a trapezoid (since $MN$ and $AB$ are parallel horizontal segments) is given by:
\[ \text{Area} = \frac{1}{2} (a + b) h \]
where $a$ and $b$ are the lengths of the parallel sides, and $h$ is the vertical height.
Step 3: Detailed Explanation:
• First, let us find the coordinates of points A and B where the parabola $y = -x^2 + 16$ intersects the $x$-axis ($y = 0$):
\[ -x^2 + 16 = 0 \implies x^2 = 16 \implies x = \pm 4 \]
Thus, the points are $A(-4, 0)$ and $B(4, 0)$.
The length of the bottom base $AB$ is:
\[ AB = 4 - (-4) = 8 \text{ units} \]
• Next, let us find the coordinates of points M and N where the parabola intersects the line $y = 7$:
\[ -x^2 + 16 = 7 \implies x^2 = 9 \implies x = \pm 3 \]
Thus, the points are $M(-3, 7)$ and $N(3, 7)$.
The length of the top base $MN$ is:
\[ MN = 3 - (-3) = 6 \text{ units} \]
• Since both segments are horizontal, they are parallel to each other.
The quadrilateral $ABNM$ is an isosceles trapezoid.
The vertical height $h$ is the distance between $y = 0$ and $y = 7$, which is:
\[ h = 7 - 0 = 7 \text{ units} \]
• Now, we can calculate the area of the trapezoid:
\[ \text{Area} = \frac{1}{2} (AB + MN) \cdot h \]
\[ \text{Area} = \frac{1}{2} (8 + 6) \cdot 7 = \frac{1}{2} \cdot 14 \cdot 7 = 49 \]
Step 4: Final Answer:
The area of the quadrilateral with vertices A, B, M, and N is 49.
The question asks for the area of the quadrilateral formed by the intersection points of a parabola with the $x$-axis and a horizontal line.
By determining the coordinates of these four points, we can identify the shape of the quadrilateral and calculate its area.
Step 2: Key Formula or Approach:
The intersection points on the $x$-axis are found by setting $y = 0$.
The intersection points with the line $y = 7$ are found by setting $y = 7$.
The area of a trapezoid (since $MN$ and $AB$ are parallel horizontal segments) is given by:
\[ \text{Area} = \frac{1}{2} (a + b) h \]
where $a$ and $b$ are the lengths of the parallel sides, and $h$ is the vertical height.
Step 3: Detailed Explanation:
• First, let us find the coordinates of points A and B where the parabola $y = -x^2 + 16$ intersects the $x$-axis ($y = 0$):
\[ -x^2 + 16 = 0 \implies x^2 = 16 \implies x = \pm 4 \]
Thus, the points are $A(-4, 0)$ and $B(4, 0)$.
The length of the bottom base $AB$ is:
\[ AB = 4 - (-4) = 8 \text{ units} \]
• Next, let us find the coordinates of points M and N where the parabola intersects the line $y = 7$:
\[ -x^2 + 16 = 7 \implies x^2 = 9 \implies x = \pm 3 \]
Thus, the points are $M(-3, 7)$ and $N(3, 7)$.
The length of the top base $MN$ is:
\[ MN = 3 - (-3) = 6 \text{ units} \]
• Since both segments are horizontal, they are parallel to each other.
The quadrilateral $ABNM$ is an isosceles trapezoid.
The vertical height $h$ is the distance between $y = 0$ and $y = 7$, which is:
\[ h = 7 - 0 = 7 \text{ units} \]
• Now, we can calculate the area of the trapezoid:
\[ \text{Area} = \frac{1}{2} (AB + MN) \cdot h \]
\[ \text{Area} = \frac{1}{2} (8 + 6) \cdot 7 = \frac{1}{2} \cdot 14 \cdot 7 = 49 \]
Step 4: Final Answer:
The area of the quadrilateral with vertices A, B, M, and N is 49.
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