If a tangent line is drawn to the parabola y = 3 - x^2 at any point on the curve in the first quadrant, a triangle is formed with the axes. At what point on the curve should the tangent be drawn to form a triangle of least are?
area*
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To determine the point on the curve where the tangent should be drawn to form a triangle of the least area, let's analyze the problem step by step.
Step 1: Find the equation of the tangent line
To find the equation of the tangent line to the parabola y = 3 - x^2, we need to find its slope and a point on the curve.
The slope of the tangent line can be determined by finding the derivative of the function y = 3 - x^2.
Taking the derivative, we get:
dy/dx = -2x
Now, let's find a point on the curve. Since we are looking for a point in the first quadrant, let's find the x-coordinate where the tangent line intersects the curve.
Setting the derivative equal to zero to find stationary points:
-2x = 0
x = 0
Substituting x = 0 into the original equation, we find the corresponding y-coordinate:
y = 3 - (0)^2 = 3
So, the point where the tangent line intersects the curve is (0, 3), which will be included in our triangle.
Now that we have the slope (-2x) and a point (0, 3) on the tangent line, we can write the equation of the tangent line using the point-slope form:
y - y1 = m(x - x1)
y - 3 = -2x(0 - 0)
y - 3 = 0
y = 3
Step 2: Find the coordinates of the triangle vertices
The triangle vertices will be (0, 3), (0, 0), and the x-intercept of the tangent line.
To find the x-intercept of the tangent line, we set y = 0 in the equation of the tangent line:
0 = 3
This equation has no real solutions, meaning the tangent line does not intersect the x-axis in the first quadrant. Therefore, there is no triangle with an x-intercept in the first quadrant.
Step 3: Analyze the triangle's area
Since we don't have an x-intercept in the first quadrant, we cannot form a triangle with the axes with a non-zero area. Therefore, there is no triangle of least area that can be formed by drawing a tangent to the parabola y = 3 - x^2 in the first quadrant.