 Questions & Answers    Question Answers

# If O is vertex of parabola and the foot of perpendicular be H from the focus S on any tangent to a parabola at any point P, then OS, OH, SP are in(A). A.P(B). G.P(C). H.P(D). A.G.P  Answer Verified
Hint: Here we first calculate the coordinates of each point and get the distance OH, OS, SP. After that we need to check the condition which one is satisfying from the option.

Complete step-by-step answer:
Let’s assume parabola ${y^2} = 4ax$
We know that parametric coordinates of any point on parabola is $\left( {a{t^2},2at} \right)$ where t is a parameter Now we can write the equation of tangent at point (P)
$ty = x + a{t^2}$

To determine the coordinate of point H, put the value of x as zero.
Therefore, the coordinate of point H is (0, at)

Now we calculate the distance between two points S and H
$SH = \sqrt {{{\left( {0 - a} \right)}^2} + {{\left( {at - 0} \right)}^2}} = \sqrt {{a^2} + {a^2}{t^2}}$

Similarly, we calculate the distance between two points O and S
$OS = \sqrt {{{\left( {0 - a} \right)}^2} + {{\left( {0 - 0} \right)}^2}} = \sqrt {{a^2}} = 0$

Similarly, we calculate the distance between two points S and P
$SP = \sqrt {{{\left( {a{t^2} - a} \right)}^2} + {{\left( {2at - 0} \right)}^2}} = a\left( {1 + {t^2}} \right)$
Now we clearly see that
$S{H^2} = OS \cdot SP$
This is the condition of G.P
So, option (b) is correct.

NOTE:
Any point on parabola ${y^2} = 4ax$ is $\left( {a{t^2},2at} \right)$ and we refer to it as the point ‘t’. Here, ‘t’ is a parameter i.e. it varies from point to point.
Bookmark added to your notes.
View Notes
Equation of Parabola  Parabola Graph  Bones of The Foot  Foot  The Story of Washing Soda Na₂CO₃ 10H₂O  What is the Formula of Tan3A?  What is the Scattering of Light?  What is the full form of AM and PM  Speech on If I Were The Prime Minister of India  What is The Full Form of PhD?  