Question
Show that the normal at a point (at^{2}, 2at) on the parabola y^{2} = 2ax cuts the curve again at the point whose parameter .




None of these
easy
Solution
The equation of the normal at the point (at^{2}, 2at) is
y – 2at_{1} = –t_{1}(x – t_{1}^{2}) … (1)
It this normal meets the parabola again the point (at^{2}, 2at), then
2at_{2} – 2at_{1} = 1 – t_{1}(at_{2}^{2} – at_{1}^{2})
or t_{1}(t_{2} + t_{1}) = –2
SIMILAR QUESTIONS
The equation represents a parabola if is
‘t_{1}’ and ‘t_{2}’ are two points on the parabola y^{2} = 4x. If the chord joining them is a normal to the parabola at ‘t_{1}’ then
The vertex of the parabola y^{2} = 8x is at the center of a circle and the parabola cuts the circle at the ends of its latus rectum. Then the equation of the circle is
Find the equation of the parabola whose focus is (1, 1) and the directrix is x + y + 1 = 0.
If the line 2x + 3y = 1 touch the parabola y^{2} = 4ax at the pointP. Find the focal distance of the point P.
Find the angle between the tangents of the parabola y^{2} = 8x, which are drawn from the point (2, 5).
Find the locus of middle point of chord y^{2} = 4ax drawn through vertex.
Find the locus of the midpoint of the chords of the parabola y^{2} = 4ax which subtend a right angle at the vertex of the parabola.
Show that the normal at a point (at^{2}, 2at) on the parabola y^{2} = 2ax cuts the curve again at the point whose parameter .
Find the locus of a pint P which moves such that two of the three normal’s drawn from it to the parabola y^{2} = 4ax are mutually perpendicular.