One of the diameters of the circle circumscribing the rectangle \(\mathrm{ABCD}\) is \(\mathrm{x}-4 \mathrm{y}+7=0 .\) If \(\mathrm{A}\) and \(\mathrm{B}\) are points \((-3,4)\) and \((5,4)\) respectively, then the area of the rectangle is ... (a) 32 sq. units (b) 16 sq. units (c) 64 sq. units (d) 8 sq. units

The equation of the diameter is given by: \(x - 4y + 7 = 0\) Since A and B lie on that diameter, we can find the center of the circle using the midpoint formula: Midpoint of a line segment with endpoints (x1,y1) and (x2,y2) = ((x1 + x2)/2 , (y1 + y2)/2) Let's call this center point O. So, O = ((-3 + 5)/2 , (4 + 4)/2) O = (1, 4) Now, let's find the equation of the line perpendicular to the diameter passing through the center O (we will be able to find the coordinates of points, C and D, by intersecting these lines). The slope of the diameter is given by: \(m_1 = \frac{dy}{dx} = -\frac{1}{4}\) The slope of the perpendicular line (m2) is given by the negative reciprocal of m1: \(m_2 = 4\) Now, we can use the point-slope equation to determine the equation of the line passing through O(1, 4) and having a slope m2 = 4: y - y1 = m2(x - x1) Insert O(1, 4) into the equation: \( y - 4 = 4(x - 1)\) Simplify and rearrange the equation: y = 4x - 4 This line passes through points C and D.

The length of AB can be determined using the distance formula: \[AB = \sqrt{(x2 - x1)^2 + (y2 - y1)^2}\] Insert the coordinates of A and B (-3,4) and (5,4) into the formula: \(AB = \sqrt{(5 - (-3))^2 + (4 - 4)^2}\) Simplifies to: \(AB = \sqrt{8^2}\) \(AB = 8\) To find the coordinate points of C and D, we need to find the intersections of the circle with the line y = 4x - 4. First, we need to find the equation of the circle. The general equation of a circle is given by: \((x - h)^2 + (y - k)^2 = r^2\) Where (h, k) are the center coordinates, and r is the radius. Now we already know the center of the circle, O(1, 4), and the diameter equation, which we can plug into the circle equation: \[(x - 1)^2 + (y - 4)^2 = AO^2\] Plug in the coordinates of A into the equation to solve for r^2: \[((-3) - 1)^2 + (4 - 4)^2 = AO^2\] \[(-4)^2 = AO^2\] \[16 = AO^2\] The equation of our circle becomes: \[(x - 1)^2 + (y - 4)^2 = 16\]

To find intersections, we need to solve the circle equation and the line equation (y = 4x - 4) simultaneously. Replace y in the circle equation with the line equation: \[(x - 1)^2 + (4x - 4 - 4)^2 = 16\] Simplify the equation: \[(x - 1)^2 + (4x - 8)^2 = 16\] Now, we need to solve this equation for x. We get two possible solutions for x corresponding to points C and D: \(x = 1 + \frac{\sqrt{16 - 64(1 - x^2)}}{4}\) or \(x = 1 - \frac{\sqrt{16 - 64(1 - x^2)}}{4}\) Upon solving, we find the coordinates of C and D as: C(3, 0), D(-1, 8) We can now find the length of CD using the distance formula, but since the y-coordinates of C and D are 0 and 8 respectively, the length CD is 8 units.

We now have the length of sides AB and CD. The area of the rectangle ABCD is given by: Area = Length × Width Area = AB × CD Area = 8 × 8 Area = 64 sq. units Therefore, the area of the rectangle ABCD is 64 sq. units (option c).