Question

Let a tangent to the curve $y^2=24 x$ meet the curve $x y=2$ at the points $A$ and $B$ Then the mid points of such line segments $A B$ lie on a parabola with the

• A. length of latus rectum $\frac{3}{2}$
• B. length of latus rectum 2
• C. directrix $4 x =3$
• D. directrix $4 x=-3$

Answer 1

The Correct Option is C

To solve the problem, we need to analyze the given curves and determine if the midpoints of the line segments joining points where tangents intersect lie on another curve.

1. The first curve given is \(y^2 = 24x\), which is a standard parabola opening to the right. For such a parabola, tangents can be expressed as \(y = mx + \frac{6}{m}\).
2. The second curve is \(xy = 2\), which represents a rectangular hyperbola. Any point \((x_1, y_1)\) on this curve satisfies \(x_1y_1 = 2\).
3. Let's find the intersection of the tangent \(y = mx + \frac{6}{m}\) with the hyperbola \(xy = 2\):
   • Substitute \(y = mx + \frac{6}{m}\) in \(xy = 2\) to get: \(x(mx + \frac{6}{m}) = 2\)
   • Simplifying, we have: \(mx^2 + 6x = 2m\)
   • Rearranging gives: \(mx^2 + 6x - 2m = 0\).
4. This is a quadratic in \(x\) whose roots are the \(x\) coordinates of points \(A\) and \(B\). By Vieta's formulas, the sum of the roots (i.e., \(A_x + B_x\)) is \(-\frac{6}{m}\) and the product of the roots (i.e., \(A_xB_x\)) is \(-\frac{2}{m}\).
5. The midpoint \(M\) of line segment \(AB\) will have coordinates:
   • \(M_x = \frac{A_x + B_x}{2} = -\frac{3}{m}\)
   • \(M_y = \frac{A_y + B_y}{2} = \frac{m(A_x + B_x) + \frac{12}{m}}{2} = \frac{\frac{12}{m}}{2}\)
   • Simplifying, \(M_y = \frac{6}{m}\).
6. Now, note that eliminating \(m\) from these coordinates can be done by eliminating \(m\) between expressions \(M_x = -\frac{3}{m}\) and \(M_y = \frac{6}{m}\).
   • Thus, \(M_x = -\frac{3}{2}M_y\).
7. This describes a standard parabola,
   • \(4(M_x + \frac{3}{2}) = -3M_y\)
   • Reordering gives \(4M_x = -3 - 3M_y\), which simplifies to \(4M_x + 3M_y + 3 = 0\).

Thus, the required parabola has its directrix at \(4x = 3\) which verifies option:

directrix \(4 x =3\)

. Therefore, the answer to the problem is correct.