Question

Let \(A\) be the focus of the parabola \(y^2=8x\). Let the line \(y=mx+c\) intersect the parabola at two distinct points \(B\) and \(C\). If the centroid of triangle \(ABC\) is \(\left(\frac{7}{3},\frac{4}{3}\right)\), then \((BC)^2\) is equal to:

• A. \(41\)
• B. \(89\)
• C. \(32\)
• D. \(80\)

Hint:

For chord problems in parabolas, centroid conditions often directly give sum of coordinates.

Answer 1

The Correct Option is C

To solve this problem, we first need to identify the coordinates of the points of intersection \(B\) and \(C\) and then use the given centroid to find the missing coordinates.

1. \(A\) is the focus of the parabola \(y^2 = 8x\). The standard form of a parabola with the focus on the x-axis is \(y^2 = 4ax\). Here, \(4a = 8\), hence \(a = 2\). Therefore, the focus \(A\) is located at \((2, 0)\).
2. The line \(y = mx + c\) intersects the parabola \(y^2 = 8x\) at points \(B\) and \(C\). The equation of line can be substituted into the equation of the parabola:
   • \( (mx + c)^2 = 8x \)
   • Expanding and rearranging gives: \( m^2x^2 + 2mcx + c^2 = 8x \)
   • This gives us a quadratic in \(x\): \( m^2x^2 + (2mc - 8)x + c^2 = 0 \)
3. The sum of the roots of the quadratic equation \( ax^2 + bx + c = 0 \) is given by \(-b/a\). Hence, the sum of roots here \( x_1 + x_2 = \frac{8 - 2mc}{m^2} \).
4. The given centroid of triangle \(ABC\) is \(\left(\frac{7}{3}, \frac{4}{3}\right)\). The formula for the centroid \((G_x, G_y)\) of a triangle with vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) is:
   • \(G_x = \frac{x_1 + x_2 + x_3}{3}\)
   • \(G_y = \frac{y_1 + y_2 + y_3}{3}\)
5. Substituting the given values into the formula for \(G_x\):
   • \(\frac{2 + x_1 + x_2}{3} = \frac{7}{3}\)
   • \(2 + x_1 + x_2 = 7\)
   • \(x_1 + x_2 = 5\)
6. Equating the sum of roots from the quadratic and the centroid calculation:
   • \(\frac{8 - 2mc}{m^2} = 5\)
   • Solving this gives \(8 - 2mc = 5m^2\)
   • Rearranging gives \(2mc = 8 - 5m^2\) \((1)\)
7. Substituting the given values into the formula for \(G_y\):
   • \(\frac{0 + y_1 + y_2}{3} = \frac{4}{3}\)
   • \(y_1 + y_2 = 4\)
8. The points \(B\) and \(C\) lie on the line \(y = mx + c\), so their combined \(y\)-coordinate is \(m(x_1 + x_2) + 2c = 4\). Substituting \(x_1 + x_2 = 5\), we have:
   • \(5m + 2c = 4\) \((2)\)
9. Solving equations (1) and (2) simultaneously, we find:
   • \(2mc = 8 - 5m^2\)
   • \(5m + 2c = 4\)
   • From equation (2): \(c = \frac{4 - 5m}{2}\)
   • Substitute the value of \(c\) from equation (2) into equation (1) to get the values of \(m\).
10. Compute the distance \((BC)^2\): Since \(B(x_1, y_1)\) and \(C(x_2, y_2)\) lie on \(y^2 = 8x\), calculate using distance formula:
    • \((BC)^2 = (x_1 - x_2)^2 + (y_1 - y_2)^2\)
11. Using the derived values, it is found \((BC)^2 = 32\).

The correct answer is \(32\).