# 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

358.8k+ views

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.

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.

Last updated date: 21st Sep 2023

•

Total views: 358.8k

•

Views today: 9.58k

Recently Updated Pages

What is the Full Form of DNA and RNA

What are the Difference Between Acute and Chronic Disease

Difference Between Communicable and Non-Communicable

What is Nutrition Explain Diff Type of Nutrition ?

What is the Function of Digestive Enzymes

What is the Full Form of 1.DPT 2.DDT 3.BCG