Identifying the Locus S
The given parabola is $y^2 = 4x$. The origin O is $(0, 0)$.
Consider a chord passing through the origin O. Since O is on the parabola, one endpoint of the chord is O$(0, 0)$. Let the other endpoint be A$(x_1, y_1)$.
Since A$(x_1, y_1)$ lies on the parabola $y^2 = 4x$, its coordinates satisfy the equation:
$ y_1^2 = 4x_1 $
Let M$(h, k)$ be the mid-point of the chord OA. Using the midpoint formula:
$ h = \frac{0 + x_1}{2} \implies x_1 = 2h $
$ k = \frac{0 + y_1}{2} \implies y_1 = 2k $
Substitute these expressions for $x_1$ and $y_1$ into the parabola equation:
$ (2k)^2 = 4(2h) $
$ 4k^2 = 8h $
$ k^2 = 2h $
Thus, the locus of the mid-point M, which is the curve S, is $y^2 = 2x$.
Finding the Locus of the Dividing Point
Let P$(h, k)$ be any point on the locus S. Therefore, P satisfies the equation:
$ k^2 = 2h $
Let Q$(x', y')$ be the point which internally divides the line segment OP in the ratio 3:1.
Using the section formula for internal division:
$ x' = \frac{1 \cdot 0 + 3 \cdot h}{3+1} = \frac{3h}{4} $
$ y' = \frac{1 \cdot 0 + 3 \cdot k}{3+1} = \frac{3k}{4} $
From these equations, we can express $h$ and $k$ in terms of $x'$ and $y'$:
$ h = \frac{4x'}{3} $
$ k = \frac{4y'}{3} $
Now, substitute these expressions for $h$ and $k$ into the equation of locus S ($k^2 = 2h$):
$ \left(\frac{4y'}{3}\right)^2 = 2\left(\frac{4x'}{3}\right) $
$ \frac{16(y')^2}{9} = \frac{8x'}{3} $
Multiply both sides by 9:
$ 16(y')^2 = 9 \times \frac{8x'}{3} $
$ 16(y')^2 = 3 \times 8x' $
$ 16(y')^2 = 24x' $
Divide both sides by 8:
$ 2(y')^2 = 3x' $
Therefore, the locus of the point Q$(x', y')$ is $2y^2 = 3x$.
Similar Questions
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Important Questions from Coordinate Geometry
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