CameraIcon

SearchIcon

MyQuestionIcon

1

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Particular Solution of a Differential Equation

If P, Q, and ...

Question

If P, Q, and R be three points on a parabola whose ordinates are in geometrical progression, prove that the tangents at P and R meet on the ordinate of Q.

Open in App

Solution

$P (a t_{1}^{2}, 2 a t_{1}), Q (a t_{2}^{2}, 2 a t_{2}), R (a t_{3}^{2}, 2 a t_{3})$
given ${(2 a t_{2})}^{2} = (2 a t_{1}) (2 a t_{3})$
$t_{2}^{2} = t_{1} t_{3} t_{2} = \sqrt{t_{1} t_{3}}$
So $Q$ is $(a t_{1} t_{3}, 2 a \sqrt{t_{1} t_{3}}) . . . . . . . (i)$
Equation of tangent at $P$ is
$t_{1} y = x + a t_{1}^{2}$
Equation of tangent at $R$ is
$t_{3} y = x + a t_{3}^{2}$
Point of intersection of tangents is $(a t_{1} t_{3}, a (t_{1} + t_{3}))$
It also lies on the parabola
$∴ {(a (t_{1} + t_{3}))}^{2} = 4 a (a t_{1} t_{3}) \Rightarrow t_{1} + t_{3} = 2 \sqrt{t_{1} t_{3}} a (t_{1} + t_{3}) = 2 a \sqrt{t_{1} t_{3}}$
The ordinate of $Q$ obtained in $(i)$ is also same
Hence proved.

Suggest Corrections

thumbs-up

0

Similar questions

P, Q, R

are three points on a parabola

y^{2} = 4 a x

whose ordinates are in geometrical progression, then the tangents at

P

R

P, Q, R

are three points on a parabola

y^{2} = 4 a x

whose ordinates are in geometric progression, then tangents at

P

R

Q. If the normals drawn from any point to the parabola cut the line

x = 2 a

in points whose ordinates are in arithmetical progression, prove that the tangents of the angles which the normals make with the axis are in geometrical progression.

Q. Ordinates of three points

A, B, C

on the parabola

y^{2} = 4 a x

are in G.P. Tangents at

A

C

Q. Ordinates of three points

A, B, C

on the parabola

y^{2} = 4 a x

G . P

A

C

View More

Join BYJU'S Learning Program

similar_icon

Related Videos

lock

General and Particular Solutions of a DE

MATHEMATICS

Watch in App

explore_more_icon

Explore more

Particular Solution of a Differential Equation

Standard XII Mathematics

Join BYJU'S Learning Program

CrossIcon