Question
The equation of a tangent to the parabola $${y^2} = 8x$$ is $$y = x + 2.$$ The point on this line from which the other tangent to the parabola is perpendicular to the given tangent is-
A.
$$\left( { 2,\,4} \right)$$
B.
$$\left( { - 2,\,0} \right)$$
C.
$$\left( { - 1,\,1} \right)$$
D.
$$\left( { 0,\,2} \right)$$
Answer :
$$\left( { - 2,\,0} \right)$$
Solution :
Parabola $${y^2} = 8x$$
We know that the locus of point of intersection of two perpendicular tangents to a parabola is its directrix.
Point must be on the directrix of parabola
$$\because $$ equation of directrix $$x + 2 = 0\,\,\, \Rightarrow x = - 2$$
Hence the point is $$\left( { - 2,\,0} \right)$$
Parabola $${y^2} = 8x$$
We know that the locus of point of intersection of two perpendicular tangents to a parabola is its directrix.
Point must be on the directrix of parabola
$$\because $$ equation of directrix $$x + 2 = 0\,\,\, \Rightarrow x = - 2$$
Hence the point is $$\left( { - 2,\,0} \right)$$