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
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
Updated On: Mar 19, 2025
- length of latus rectum $\frac{3}{2}$
- length of latus rectum 2
- directrix $4 x =3$
- directrix $4 x=-3$
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The Correct Option is C
Approach Solution - 1
The correct answer is (C) : directrix $4 x =3$
.......(1)
.........(2)
From (1) and (2)
then locus is
Therefore directrix is
.......(1)
.........(2)
From (1) and (2)
then locus is
Therefore directrix is
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Approach Solution -2
The given parabola is \( y^2 = 24x \). Let \( a = 6 \).
From the equations of tangency:
\[
AB = xy = x + 6t^2 \tag{1}
\]
\[
AB = t \Rightarrow kx + hy = 2hk \tag{2}
\]
From (1) and (2), we derive:
\[
k = \frac{h}{1 - t}, \quad h = \frac{2hk}{1 - 6t^2}.
\]
Simplifying the locus, we get:
\[
y^2 = -3x.
\]
Thus, the directrix is \( 4x = 3 \).
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Concepts Used:
Parabola
Parabola is defined as the locus of points equidistant from a fixed point (called focus) and a fixed-line (called directrix).
Standard Equation of a Parabola
For horizontal parabola
- Let us consider
- Origin (0,0) as the parabola's vertex A,
- Two equidistant points S(a,0) as focus, and Z(- a,0) as a directrix point,
- P(x,y) as the moving point.
- Let us now draw SZ perpendicular from S to the directrix. Then, SZ will be the axis of the parabola.
- The centre point of SZ i.e. A will now lie on the locus of P, i.e. AS = AZ.
- The x-axis will be along the line AS, and the y-axis will be along the perpendicular to AS at A, as in the figure.
- By definition PM = PS
=> MP2 = PS2
- So, (a + x)2 = (x - a)2 + y2.
- Hence, we can get the equation of horizontal parabola as y2 = 4ax.
For vertical parabola
- Let us consider
- Origin (0,0) as the parabola's vertex A
- Two equidistant points, S(0,b) as focus and Z(0, -b) as a directrix point
- P(x,y) as any moving point
- Let us now draw a perpendicular SZ from S to the directrix.
- Then SZ will be the axis of the parabola. Now, the midpoint of SZ i.e. A, will lie on P’s locus i.e. AS=AZ.
- The y-axis will be along the line AS, and the x-axis will be perpendicular to AS at A, as shown in the figure.
- By definition PM = PS
=> MP2 = PS2
So, (b + y)2 = (y - b)2 + x2
- As a result, the vertical parabola equation is x2= 4by.