If a line along a chord of the circle \(4x^2 + 4y^2 + 120x + 675 = 0\), passes through the point \((-30, 0)\) and is tangent to the parabola \(y^2 = 30x\), then the length of this chord is :
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For any circle-chord problem, drawing the triangle formed by the radius, the perpendicular distance from the center, and half the chord length is the standard geometric approach.
Step 1: Understanding the Concept:
First, we find the equation of the tangent to the parabola that passes through the given point. This tangent acts as a chord for the given circle. Then, we use the radius and the distance from the center to the chord to calculate the chord's length. Step 2: Key Formula or Approach:
1. Tangent to \(y^2 = 4ax\) is \(y = mx + a/m\).
2. Circle \(x^2 + y^2 + 2gx + 2fy + c = 0\) has center \((-g, -f)\) and radius \(r = \sqrt{g^2+f^2-c}\).
3. Chord length = \(2\sqrt{r^2 - d^2}\), where \(d\) is the distance from center to chord. Step 3: Detailed Explanation:
For the parabola \(y^2 = 30x\), \(4a = 30 \implies a = 7.5\).
Tangent: \(y = mx + 7.5/m\).
It passes through \((-30, 0)\):
\[ 0 = -30m + 7.5/m \implies 30m^2 = 7.5 \implies m^2 = 1/4 \implies m = \pm 1/2 \]
Taking \(m = 1/2\), the line is \(y = \frac{1}{2}x + 15 \implies x - 2y + 30 = 0\).
Now for the circle: \(x^2 + y^2 + 30x + 168.75 = 0\).
Center \(C(-15, 0)\) and \(r^2 = (-15)^2 - 168.75 = 225 - 168.75 = 56.25 \implies r = 7.5\).
Distance \(d\) from \(C(-15, 0)\) to \(x - 2y + 30 = 0\):
\[ d = \frac{|-15 - 0 + 30|}{\sqrt{1^2 + (-2)^2}} = \frac{15}{\sqrt{5}} = 3\sqrt{5} \]
Chord length:
\[ L = 2\sqrt{r^2 - d^2} = 2\sqrt{56.25 - 45} = 2\sqrt{11.25} = 2\sqrt{\frac{45}{4}} = \sqrt{45} = 3\sqrt{5} \] Step 4: Final Answer:
The length of the chord is \(3\sqrt{5}\).
