Three normals to y2 = 4x pass through the point (15,12) Show that one of the normals is given - Maths - Conic Sections

Question

Three normals to y2 = 4x pass through the point (15,12) Show
that one of the normals is given by y = x - 3 find the equations of
the others

Himanshu · Accepted Answer

Consider a point having co-ordinates (t2, 2t) Equation of normal to the parabola y2 = 4x at point (t2, 2t) is given by $y = 2 t + t^{3} - t x$ $12 = 2 t + t^{3} - 15 t \Rightarrow t^{3} - 13 t + 12 = 0 \Rightarrow (t - 1) (t^{2} - 12 - t) = 0 \Rightarrow (t - 1) (t^{2} - 4 t + 3 t - 12) = 0 \Rightarrow (t - 1) (t - 4) (t + 3) = 0 \Rightarrow t = 1 o r 4 o r - 3 y = 2 (4) + 4^{3} - 4 x \Rightarrow y + 4 x - 72 = 0 y = 2 (- 3) + (- 3)^{3} + 3 x \Rightarrow y - 3 x + 33 = 0$

Three normals to y2 = 4x pass through the point (15,12). Show that one of the normals is given by y = x - 3 find the equations of the others.