To find the limits of m for which the line 3x - 4y = m cuts the circle x^2 + y^2 - 4x - 8y - 5 = 0 in two points, we can follow these steps:

Step 1: Rewrite the equation of the circle in standard form: x^2 - 4x + y^2 - 8y = 5

Step 2: Complete the square for both x and y terms: (x^2 - 4x + 4) + (y^2 - 8y + 16) = 5 + 4 + 16 (x - 2)^2 + (y - 4)^2 = 25

Step 3: Compare the equation of the circle with the general equation of a circle: (x - h)^2 + (y - k)^2 = r^2

From the comparison, we can determine the center of the circle as (h, k) = (2, 4) and the radius as r = 5.

Step 4: Substitute the equation of the line into the equation of the circle: 3x - 4y = m x = (m + 4y) / 3

Substituting x into the equation of the circle: ((m + 4y) / 3 - 2)^2 + (y - 4)^2 = 25

Step 5: Simplify the equation and solve for y: (m^2 + 8my + 16y^2 + 9 - 12m + 36) / 9 + (y - 4)^2 = 25 (m^2 + 8my + 16y^2 - 12m + 20) / 9 + (y - 4)^2 = 25 (m^2 + 8my + 16y^2 - 12m + 20) / 9 = 25 - (y - 4)^2

Step 6: Determine the conditions for the line to intersect the circle in two points: For the line to intersect the circle in two points, the discriminant of the equation must be greater than zero.

The discriminant is given by: D = (8m)^2 - 4(16)(m^2 + 20 - 9(25 - (y - 4)^2)) D = 64m^2 - 64(m^2 + 20 - 9(25 - (y - 4)^2)) D = 64m^2 - 64m^2 - 1280 + 576(25 - (y - 4)^2) D = 576(25 - (y - 4)^2) - 1280

For the line to intersect the circle in two points, D > 0: 576(25 - (y - 4)^2) - 1280 > 0 576(25 - (y - 4)^2) > 1280 25 - (y - 4)^2 > 1280 / 576 25 - (y - 4)^2 > 2.222

Step 7: Solve for y: (y - 4)^2 < 25 - 2.222 (y - 4)^2 < 22.778 -√22.778 < y - 4 < √22.778 4 - √22.778 < y < 4 + √22.778

Step 8: Substitute the limits of y back into the equation of the line to find the corresponding limits of m: For y = 4 - √22.778: 3x - 4(4 - √22.778) = m 3x - 16 + 4√22.778 = m 3x = m + 16 - 4√22.778 x = (m + 16 - 4√22.778) / 3

For y = 4 + √22.778: 3x - 4(4 + √22.778) = m 3x - 16 - 4√22.778 = m 3x = m + 16 + 4√22.778 x = (m + 16 + 4√22.778) / 3

Therefore, the limits of m for which the line 3x - 4y = m cuts the circle x^2 + y^2 - 4x - 8y - 5 = 0 in two points are: m > (m + 16 - 4√22.778) / 3 m < (m + 16 + 4√22.778) / 3

Question

To find the limits of m for which the line 3x - 4y = m cuts the circle x^2 + y^2 - 4x - 8y - 5 = 0 in two points, we can follow these steps:

Step 1: Rewrite the equation of the circle in standard form: x^2 - 4x + y^2 - 8y = 5

Step 2: Complete the square for both x and y terms: (x^2 - 4x + 4) + (y^2 - 8y + 16) = 5 + 4 + 16 (x - 2)^2 + (y - 4)^2 = 25

Step 3: Compare the equation of the circle with the general equation of a circle: (x - h)^2 + (y - k)^2 = r^2

From the comparison, we can determine the center of the circle as (h, k) = (2, 4) and the radius as r = 5.

Step 4: Substitute the equation of the line into the equation of the circle: 3x - 4y = m x = (m + 4y) / 3

Substituting x into the equation of the circle: ((m + 4y) / 3 - 2)^2 + (y - 4)^2 = 25

Step 5: Simplify the equation and solve for y: (m^2 + 8my + 16y^2 + 9 - 12m + 36) / 9 + (y - 4)^2 = 25 (m^2 + 8my + 16y^2 - 12m + 20) / 9 + (y - 4)^2 = 25 (m^2 + 8my + 16y^2 - 12m + 20) / 9 = 25 - (y - 4)^2

Step 6: Determine the conditions for the line to intersect the circle in two points: For the line to intersect the circle in two points, the discriminant of the equation must be greater than zero.

The discriminant is given by: D = (8m)^2 - 4(16)(m^2 + 20 - 9(25 - (y - 4)^2)) D = 64m^2 - 64(m^2 + 20 - 9(25 - (y - 4)^2)) D = 64m^2 - 64m^2 - 1280 + 576(25 - (y - 4)^2) D = 576(25 - (y - 4)^2) - 1280

For the line to intersect the circle in two points, D > 0: 576(25 - (y - 4)^2) - 1280 > 0 576(25 - (y - 4)^2) > 1280 25 - (y - 4)^2 > 1280 / 576 25 - (y - 4)^2 > 2.222

Step 7: Solve for y: (y - 4)^2 < 25 - 2.222 (y - 4)^2 < 22.778 -√22.778 < y - 4 < √22.778 4 - √22.778 < y < 4 + √22.778

Step 8: Substitute the limits of y back into the equation of the line to find the corresponding limits of m: For y = 4 - √22.778: 3x - 4(4 - √22.778) = m 3x - 16 + 4√22.778 = m 3x = m + 16 - 4√22.778 x = (m + 16 - 4√22.778) / 3

For y = 4 + √22.778: 3x - 4(4 + √22.778) = m 3x - 16 - 4√22.778 = m 3x = m + 16 + 4√22.778 x = (m + 16 + 4√22.778) / 3

Therefore, the limits of m for which the line 3x - 4y = m cuts the circle x^2 + y^2 - 4x - 8y - 5 = 0 in two points are: m > (m + 16 - 4√22.778) / 3 m < (m + 16 + 4√22.778) / 3

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