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If a circle whose center is (1,-3) touch...

If a circle whose center is `(1,-3)` touches the line `3x-4y-5=0` , then find its radius.

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To find the radius of a circle whose center is at the point (1, -3) and that touches the line given by the equation \(3x - 4y - 5 = 0\), we will follow these steps: ### Step 1: Identify the center of the circle and the line equation The center of the circle is given as \(C(1, -3)\) and the line equation is \(3x - 4y - 5 = 0\). ### Step 2: Use the formula for the distance from a point to a line The distance \(d\) from a point \((x_1, y_1)\) to a line \(Ax + By + C = 0\) is given by the formula: \[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \] In our case, \(A = 3\), \(B = -4\), \(C = -5\), and the point is \((x_1, y_1) = (1, -3)\). ### Step 3: Substitute the values into the distance formula Substituting the values into the formula: \[ d = \frac{|3(1) + (-4)(-3) - 5|}{\sqrt{3^2 + (-4)^2}} \] ### Step 4: Calculate the numerator Calculating the numerator: \[ 3(1) + (-4)(-3) - 5 = 3 + 12 - 5 = 10 \] Thus, the absolute value is \(|10| = 10\). ### Step 5: Calculate the denominator Calculating the denominator: \[ \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \] ### Step 6: Calculate the distance Now substituting back into the distance formula: \[ d = \frac{10}{5} = 2 \] ### Conclusion Since the distance \(d\) we calculated is the radius of the circle, the radius of the circle is \(2\) units. ### Final Answer The radius of the circle is \(2\) units. ---

To find the radius of a circle whose center is at the point (1, -3) and that touches the line given by the equation \(3x - 4y - 5 = 0\), we will follow these steps: ### Step 1: Identify the center of the circle and the line equation The center of the circle is given as \(C(1, -3)\) and the line equation is \(3x - 4y - 5 = 0\). ### Step 2: Use the formula for the distance from a point to a line The distance \(d\) from a point \((x_1, y_1)\) to a line \(Ax + By + C = 0\) is given by the formula: \[ ...
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Knowledge Check

  • The equation of the circle whose center is (3,-2) and which touches the line 3x - 4y + 13 = 0 is

    A
    A. `x^(2) + y^(2) + 6x - 4y - 23 = 0 `
    B
    B.` x^(2) + y^(2) - 6x - 4y - 23 = 0 `
    C
    C. ` x^(2) + y^(2) - 6x + 4y - 23 = 0 `
    D
    D. ` x^(2) + y^(2) + 6x + 4y + 23 = 0 `
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