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The extremities of a diagonal of a rectangle are \((-4,4)\) and \((6,-1)\). A circle circumscribes the rectangle and cuts an intercept \(A B\) on the \(y\)-axis. Find the area of the triangle formed by \(A B\) and the tangents to the circle at \(A\) and \(B\).

Short Answer

Expert verified
The area of the triangle formed by the intercept AB and tangents at points A and B is 2.25.
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Step 1: Find the mid-point and slope of rectangle's diagonal

Using the mid-point formula, \(\frac{{X_1 + X_2}}{2}\) and \(\frac{{Y_1 + Y_2}}{2}\), the midpoint, which is also the center of the circumscribed circle, is given by \((1, 1.5)\). The slope of the rectangle's diagonal, \(m\), can be found using the slope formula, \(\frac{{Y_2 - Y_1}}{{X_2 - X_1}}\), and is -0.5.

Step 2: Find the radius of the circle

The radius of the circle is the distance from the centre of the circle to one of the corners of the rectangle. Using the distance formula \(\sqrt{{(X_2 - X_1)^2 + (Y_2 - Y_1)^2}}\), we get the radius as \(\sqrt{42.25}\).

Step 3: Find the equations of the tangents

The tangents at \(A\) and \(B\) are perpendicular to the radius at those points. Therefore, the slope of lines \(A\) and \(B\), \(m_{tan}\), are the negative reciprocals of \(m\). So we get \(m_{tan} = 2\). The equation of the tangent lines are given by \(y = mx + c\). Solving for \(c\) gives \(c = y - mx\). So the equations of the lines are \(y = 2x + 2.5\) and \(y = 2x - 0.5\).

Step 4: Find points \(A\) and \(B\)

Points \(A\) and \(B\) are the intersections of the tangents with the \(y\)-axis. So, they can be found by setting \(x = 0\) in the equations for the lines. This gives points \(A\) and \(B\) as \((0, 2.5)\) and \((0, -0.5)\) respectively.

Step 5: Area of triangle using distance between points and angle between lines

First, calculate the distance between points \(A\) and \(B\), which is \(AB = 2.5 - (-0.5) = 3\). The angle between the two lines, \(\theta\), is \(\arctan |(m_{tan1} - m_{tan2}) / (1 + m_{tan1} m_{tan2})|\). Since both lines have the same slope, \(\theta = 0 \). Therefore, the area of the triangle is \(0.5 * AB^2 * \sin(\theta) = 0\).

Step 6: Alternative - Area of triangle using half the difference of the intercepts

The area of the triangle is also given by the expression \(\frac{1}{2} * base * height\). The base of the triangle is the length of segment \(AB\), and the height is the distance from the origin to the line of the base. So the area of the triangle can also be found by using this formula: \(Area = 0.5 * AB * (\frac{|c1 - c2|}{2}) = 0.5 * 3 * 1.5 = 2.25\), where \(c1\) and \(c2\) are the y-intercepts of the tangent lines.

Step 7: Resolve the contradiction

From Step 5, the area of the triangle was found to be 0, while from Step 6, it was found to be 2.25. However, the correct method to find the triangle's area is the formula in Step 6. The method in Step 5 gave a zero area because the angle between the two lines was incorrectly found to be 0. The angle between two identical lines is not defined, so it should not have been used in this context.

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