Question

If the tangent at the point $P({x_1},{y_1})$ to the parabola ${y^2} = 4ax$ meets the ${y^2} = 4a(x + b)$parabola at Q & R, then the mid-point of QR is
A. $({x_1} + b,{y_1} + b)$
B. $({x_1} - b,{y_1} - b)$
C. $({x_1},{y_1})$
D. $({x_1} + b,{y_1})$

Answer 1

The equation of tangent to the parabola ${y^2} = 4ax$ at given point $P({x_1},{y_1})$can be given by$y{y_1} = 4a(\dfrac{{x + {x_1}}}{2}) = 2a(x + {x_1})$
Using the concept mentioned and also writing the equation of for the midpoint as $T = {S_1}$.Solve both the equation as they both represent the same line and so our required answer will be obtained.

Complete step by step answer:

The tangent to the first parabola will be the chord for another parabola and hence we write the equation of tangent for the first curve using given points and we can write the equation of chord for the second curve using variable points and consider the variable point as the mid-point of the chord.
Writing the equation of tangent for given parabola is $y{y_1} = 4a(\dfrac{{x + {x_1}}}{2}) = 2a(x + {x_1})$
Now, writing the equation of chord using above given concept and let $\left( {h,{\text{ }}k} \right)$ be the given midpoint of QR,
So, the equation of QR is

$$\Rightarrow ky - 2a(x + h) - 4ab = {k^2} - 4a(h + b) \\\ \Rightarrow - 2ax + ky + 2ah - {k^2} = 0 \\\$$

Now, as both the lines are the same we just need to compare the coefficient in order to obtain the value of $\left( {h,{\text{ }}k} \right)$ and as they are the mid-point of the line so it will directly be our required answer.

$$\dfrac{{2a}}{{ - 2a}} = - \dfrac{{{y_1}}}{k} = \dfrac{{2a{x_1}}}{{2ah - {k^2}}} \\\ so, \\\ {y_1} = k \\\ {x_1} = \dfrac{{{k^2} - 2ah}}{{2a}} \\\$$

Now, further simplifying

$${x_1} = \dfrac{{{k^2} - 2ah}}{{2a}} \\\ 2a{x_1} = {y_1}^2 - 2ah \\\ u\sin g, \\\ {y_1}^2 = 2a{x_1} \\\ \Rightarrow {x_1} = h \\\$$

Hence , $\left( {h{\text{ }},{\text{ }}k} \right)$ are $({x_1},{y_1})$ and so option ( c ) is correct answer.

Note: You should remember the formula of the equation of a tangent to the parabola at the given point and apply it to find the equation of tangent, proper substitution and simplification should be done. A secant of a parabola is a line, or line segment, that joins two distinct points on the parabola. A tangent is a line that touches the parabola at exactly one point. The chord joining the points of contact of two tangents drawn from an external point to a parabola is called the chord of contact w.r.t. the given point.