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Chapter 15: Problem 1362

The equation of circle touching the axis of \(\mathrm{y}\) at a distance \(+4\) from the origin and cutoff an intercept 6 from the axis of \(\mathrm{x}\) is \(\ldots \ldots \ldots\) (a) \(x^{2}+y^{2}-10 x-8 y+16=0\) (b) \(x^{2}+y^{2}+10 x-8 y+16=0\) (c) \(x^{2}+y^{2}-10 x+8 y+16=0\) (d) none of these

Short Answer

Expert verified
The equation of the circle touching the axis of y at a distance +4 from the origin and cutting off an intercept 6 from the axis of x is \(x^2 + y^2 - 12x - 8y + 36 = 0\), which is not among the given options, so the correct answer is (d) none of these.

Step by step solution

01

Find the center of the circle

Since the circle touches the y-axis at a distance of +4 from the origin, the center of the circle must be at a point \((a, b)\) where \(b = 4\). Also, since it cuts off an intercept of 6 units from the x-axis, the radius, r, of the circle must be such that the distance between the center and the x-axis is half of the intercept, which is 3.
02

Determine the radius of the circle

Since the circle touches the y-axis at \(y = 4\), the radius of the circle must be equal to the distance between the x-axis and the center of the circle, which is 4 units. Therefore, the radius, r, is 4.
03

Determine the x-coordinate of the center

Now that we know the radius is 4 and the center of the circle has a y-coordinate of 4, we can use the distance between the center \((a, 4)\) and the x-axis to determine the x-coordinate of the center. Since the distance between the center of the circle and the x-axis is half of the x-intercept (3 units), we can say that the x-coordinate of the center is \(a = 3\cdot2 = 6\).
04

Write the equation of the circle

Now that we have the center \((6, 4)\) and the radius (4) of the circle, we can write its equation: \((x-6)^2 + (y-4)^2 = 4^2\) Expanding and simplifying the equation, we get: \(x^2 - 12x + 36 + y^2 - 8y + 16 = 16\) Combining the terms and moving constants to the right side: \(x^2 + y^2 - 12x - 8y + 36 = 0\) The correct equation for the circle is not among the given options, so the answer is: (d) none of these

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