Question

Find the locus of the middle points of chords of the parabola which pass through the fixed point $ (h,k) $ .

Answer 1

Hint: A parabola is the curve in which every point is at an equal distance from the fixed point known as the focus and the fixed straight line known as the Directrix. Here use the standard chord equation for the parabola and place the given coordinates in it for the required answer.

Complete step-by-step answer:
Let us consider that the midpoint of the chord be $ = (r,s) $
Equation of the chord can be given by, $ T = S(r,s) $
The equation of the chord becomes –
 $ {s^2} - 4ar = ys - 2ar - 2ax $
Given that the locus of the middle points of chords of the parabola pass through the fixed point $ (h,k) $
Therefore, place $ (h,k) $ in the above equation-
 $ {s^2} - 4ar = ks - 2ar - 2ah $
Simplify the above equation, when any term is moved from one side to another the sign of the term also changes positive to negative and negative to positive.
\[
\Rightarrow {s^2} = ks - 2ar - 2ah + 4ar \\
\Rightarrow {s^2} = ks\underline { - 2ar + 4ar} - 2ah \\
\Rightarrow {s^2} = ks + 2ar - 2ah \\
 \]
Rearranging the above equation –
\[{s^2} = ks + 2a(r - h)\]
Now, the equation can be written in the form of $ (x,y) $
\[
\Rightarrow {y^2} = ky + 2a(k - h) \\
\Rightarrow {y^2} = 2a(k - h) + ky \\
 \]
Hence, the required answer – the equation of the locus passing from the fixed point $ (h,k) $ is \[{y^2} = 2a(k - h) + ky\]

Note: Remember all the concepts and differences between parabola and the hyperbola and apply equations accordingly reading the question twice.
The parabola can be defined as the locus of the point which moves and as a result ultimately the same distance from the point called focus and the given line is called the Directrix. The word locus means the set of points satisfying the given specific conditions.