Question

Find the locus of point P such that chord of contact on point P with respect to \[{y^2} = 4ax\] touches the hyperbola ${x^2} - {y^2} = {a^2}$.

Answer 1

Hint: The chord which intersects two tangents drawn from two points from the parabola to a point outside the circle is known as chord of contact. For this particular question we calculate the equation of chord of contact and the equation of tangent of circle. These two equations are then compared to find the locus of point P.

Complete step by step solution:
Let point P be $({x_1},{y_1})$.
We know that the equation of chord of contact on point P with respect to \[{y^2} = 4ax\] is
$
 y{y_1} = 2a\left( {x + {x_1}} \right) \\
 y = \dfrac{{2ax}}{{{y_1}}} + \dfrac{{2a{x_1}}}{{{y_1}}} - (i) \\
 $
Hence, the slope of the line is
$m = \dfrac{{2a}}{{{y_1}}} - (ii)$
We also know that equation of a tangent of hyperbola $\dfrac{{{x^2}}}{{{a^2}}} - \dfrac{{{y^2}}}{{{b^2}}} = 1$ is
$y = mx \pm \sqrt {{a^2}{m^2} - {b^2}} - (iii)$
When we compare this equation with the hyperbola ${x^2} - {y^2} = {a^2}$, the we see that
$
 {a^2} = {b^2} \\
 a = b - (iv) \\
 $
Now, we compare $(i)$ and $(iii)$ equation,
$\dfrac{{2ax}}{{{y_1}}} + \dfrac{{2a{x_1}}}{{{y_1}}} = mx \pm \sqrt {{a^2}{m^2} - {b^2}} $
Substituting the value of $m$ from $(ii)$, we get
$
\dfrac{{2ax}}{{{y_1}}} + \dfrac{{2a{x_1}}}{{{y_1}}} = \dfrac{{2a}}{{{y_1}}}x \pm \sqrt {{a^2}{{\dfrac{{(2a)}}{{{{({y_1})}^2}}}}^2} - {b^2}} \\
 \dfrac{{2a{x}}}{{{y_1}}} + \dfrac{{2a{x_1}}}{{{y_1}}} = \dfrac{{2a{x}}}{{{y_1}}} \pm \sqrt {{a^2}{{\dfrac{{(2a)}}{{{{({y_1})}^2}}}}^2} - {b^2}} \\
 \dfrac{{2a{x_1}}}{{{y_1}}} = \sqrt {{a^2}{{\dfrac{{(2a)}}{{{{({y_1})}^2}}}}^2} - {b^2}} \\
 $
Squaring both sides we get
$
 {\left( {\dfrac{{2a{x_1}}}{{{y_1}}}} \right)^2} = {\left( {\sqrt {{a^2}{{\dfrac{{(2a)}}{{{{({y_1})}^2}}}}^2} - {b^2}} } \right)^2} \\
 \dfrac{{4{a^2}x_1^2}}{{{y}_1^2}} = \dfrac{{4{a^4} - y_1^2{b^2}}}{{{y}_1^2}} \\
 $
Substituting the value from (iv)
\[
 4{a^2}x_1^2 = 4{a^4} - y_1^2{a^2} \\
 4{{{a}}^2}x_1^2 = {{{a}}^2}(4{a^2} - y_1^2) \\
 4x_1^2 + y_1^2 = 4{a^2} \\
\]
Hence, the locus of the point P is \[4x_{}^2 + y_{}^2 = 4{a^2}\] which represents an ellipse.

Note:
We can also calculate the length of the chord of contact by drawing a perpendicular from the centre to the tangent and calculating the perpendicular distance between the lines.