If the lines 3x − 4y + 4 = 0 and 6x − 8y − 7 = 0 are the tangents to a circle, then the radius of the circle is ________.
The question asks for the radius of a circle given two tangent lines to it. The two lines are given by the equations 3x − 4y + 4 = 0 and 6x − 8y − 7 = 0.
Analyzing the Given Tangent Lines
Let's look at the equations of the two lines:
- Line 1: \(3x - 4y + 4 = 0\)
- Line 2: \(6x - 8y - 7 = 0\)
We need to determine the relationship between these two lines. Let's compare the coefficients of x and y.
For Line 1, the coefficients are A\(_1\) = 3 and B\(_1\) = -4.
For Line 2, the coefficients are A\(_2\) = 6 and B\(_2\) = -8.
Let's check the ratio of the coefficients:
\[ \frac{A_1}{A_2} = \frac{3}{6} = \frac{1}{2} \] \[ \frac{B_1}{B_2} = \frac{-4}{-8} = \frac{1}{2} \]Since \(\frac{A_1}{A_2} = \frac{B_1}{B_2}\), the two lines are parallel. These two parallel lines are tangents to the circle.
Relating Parallel Tangents to Circle Radius
When two parallel lines are tangents to a circle, the distance between these two parallel lines is equal to the diameter of the circle. Let 'd' be the distance between the lines and 'r' be the radius of the circle.
\[ d = \text{Diameter} = 2r \] \[ r = \frac{d}{2} \]So, we need to find the distance between the parallel lines \(3x - 4y + 4 = 0\) and \(6x - 8y - 7 = 0\).
Calculating the Distance Between Parallel Lines
The formula for the distance between two parallel lines \(Ax + By + C_1 = 0\) and \(Ax + By + C_2 = 0\) is:
\[ d = \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}} \]To use this formula, the coefficients of x and y (A and B) must be the same in both equations. The first line is \(3x - 4y + 4 = 0\). The second line is \(6x - 8y - 7 = 0\). We can divide the second equation by 2 to match the coefficients of the first line:
\[ \frac{6x - 8y - 7}{2} = \frac{0}{2} \] \[ 3x - 4y - \frac{7}{2} = 0 \]Now we have the two parallel lines in the form \(Ax + By + C = 0\):
- Line 1: \(3x - 4y + 4 = 0\) → \(A=3, B=-4, C_1=4\)
- Line 2: \(3x - 4y - \frac{7}{2} = 0\) → \(A=3, B=-4, C_2=-\frac{7}{2}\)
Now, we can calculate the distance 'd' between these lines:
\[ d = \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}} \] \[ d = \frac{|4 - (-\frac{7}{2})|}{\sqrt{3^2 + (-4)^2}} \] \[ d = \frac{|4 + \frac{7}{2}|}{\sqrt{9 + 16}} \] \[ d = \frac{|\frac{8}{2} + \frac{7}{2}|}{\sqrt{25}} \] \[ d = \frac{|\frac{15}{2}|}{5} \] \[ d = \frac{\frac{15}{2}}{5} \] \[ d = \frac{15}{2 \times 5} = \frac{15}{10} = \frac{3}{2} \]The distance between the two parallel tangent lines is \(\frac{3}{2}\). This distance is the diameter of the circle.
Finding the Radius of the Circle
The radius 'r' is half of the diameter:
\[ r = \frac{d}{2} \] \[ r = \frac{\frac{3}{2}}{2} \] \[ r = \frac{3}{2 \times 2} = \frac{3}{4} \]Thus, the radius of the circle is \(\frac{3}{4}\).
Let's summarise the steps:
- Identify the given lines and check if they are parallel.
- If parallel, understand that the distance between them equals the circle's diameter.
- Rewrite one line equation if necessary so that coefficients of x and y match.
- Use the distance formula for parallel lines to find the diameter.
- Calculate the radius by dividing the diameter by 2.
Revision Table: Key Concepts
| Concept | Description | Formula/Property |
|---|---|---|
| Parallel Lines | Two lines \(A_1x + B_1y + C_1 = 0\) and \(A_2x + B_2y + C_2 = 0\) are parallel. | \(\frac{A_1}{A_2} = \frac{B_1}{B_2}\) |
| Parallel Tangents | Two parallel lines tangent to a circle. | Distance between lines = Diameter of the circle. |
| Distance between Parallel Lines | Distance between \(Ax + By + C_1 = 0\) and \(Ax + By + C_2 = 0\). | \(\frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}}\) |
| Radius and Diameter | Relationship between circle's radius (r) and diameter (d). | \(d = 2r\) or \(r = \frac{d}{2}\) |
Additional Information: Circle Properties
- A tangent to a circle is a line that touches the circle at exactly one point.
- The radius drawn to the point of tangency is perpendicular to the tangent line.
- If two parallel lines are tangents, the center of the circle lies exactly midway between these two lines.
- The general equation of a circle with center \((h, k)\) and radius \(r\) is \((x - h)^2 + (y - k)^2 = r^2\).
- The general equation of a circle is \(x^2 + y^2 + 2gx + 2fy + c = 0\), where the center is \((-g, -f)\) and the radius is \(\sqrt{g^2 + f^2 - c}\).
- In this problem, we didn't need the center, only the distance between the parallel tangents to find the radius.
Important Questions from Circles
The internal center of similitude of two circles $ (x - 1)^2 + (y - 3)^2 = 4 $ and $ (x + 5)^2 + (y - 9)^2 = 16 $ is
The area of a circle is 15400 cm2. What is the positive difference between the radius and the circumference of the circle? [Use π = \(\frac{22}{7}\)]
A is a point outside of a circle with centre O. AP and AQ are two tangents of the circle. If AP = a2 + 14 and AQ = 239, then what is the value of a ?
A circle of radius 5 units touches the Co-ordinate axes in the first quadrant. If the circle makes one complete roll on x-axis along the positive direction of x-axis, find its equation in new position.
The tangent at a point C of a circle and diameter AB when extended intersect at D, if ∠DCA = 110°, then ∠CBA is equal to
