Let us substitute x = at^{2} and y = 2at in the equation of a parabola y^{2} = 4ax.
(2at)^{2} |
= |
4a.at^{2} |
4 a^{2} t^{2} |
= |
4 a^{2} t^{2} |
i.e., the two co-ordinates satisfy the equation for all real values of 't'.
Conversely, if P(x, y) is any point on y^{2} = 4ax, there exists a t ∈ R such that x = at^{2} and y = 2at.
These two are know as the parametric equations of a parabola.
The point P(at^{2}, 2at) is generally denoted by P(t) or 't'.
The S-notation
Let (x_{1}, y_{1}) and (x_{2}, y_{2}) be two points on the parabola y^{2} = 4ax, then
i) |
S |
= |
y^{2} – 4ax |
ii) |
S_{1} |
= |
yy_{1} – 2a (x + x_{1}) |
iii) |
S_{12} |
= |
y_{1}y_{2} – 2a (x_{1} + x_{2}) |
iv) |
S_{11} |
= |
y_{1}^{2} – 4ax_{1} |
Point of intersection of tangents at any two points on the parabola
(i) Let P (at^{2}_{1}, 2at_{1}) and Q (at^{2}_{2}, 2at_{2}) be two points on the parabola y^{2} = 4ax. The point of intersection of tangents at two points 'P' and 'Q' is R (at_{1}t_{2}, a(t_{1} + t_{2})).
Refer fig.(a).
(ii) The locus of the point of intersection of tangents to the parabola y^{2} = 4ax which meet at an angle 'α' is (x + a)^{2} tan^{2} α = y^{2} – 4ax.
(iii) The area of the triangle PQR (from the above fig.) is 1/2 a^{2} (t_{1} – t_{2})^{3}.
(iv) The locus of the point of intersection of perpendicular tangents to a conic is known as its director circle. The director circle of a parabola is its directrix.
Point of intersection of normals at any two points on the parabola
Let P (at_{1}^{2}, 2at_{1}) and Q (at_{2}^{2}, 2at_{2}) be two points on the parabola y^{2} = 4ax. Then the point of intersection of normals at P and Q is R (2a + a (t_{1}^{2} + t_{2}^{2} + t_{1}t_{2}), – a t_{1}t_{2} (t_{1} + t_{2})).
Refer fig.(b).
Relation between 't_{1}' and 't_{2}' if normal at 't_{1}' meets the parabola again at 't_{2}'
Let P (at_{1}^{2}, 2at_{1}) and Q (at_{2}^{2}, 2at_{2}) be two points on the parabola y^{2} = 4ax. If the normal at the point P meets the parabola again at Q then .
Refer fig.(c).