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

Byju's Answer

Standard XII

Mathematics

Equation of Parabola When Its Axis Is Parallel to X or Y Axis

Any chord AB ...

Question

Any chord $A B$ of $y^{2} = 4 a x$ cuts the axis of the parabola at $P$ , so that $A P : A B = 1 : 3$ where $A (a t_{1}^{2}, 2 a t_{1}), B (a t_{2}^{2}, 2 a t_{2})$ , then

t_{1} t_{2} = - 4

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

t_{1} + t_{2} = 0

No worries! We‘ve got your back. Try BYJU‘S free classes today!

2 t_{1} + t_{2} = 0

No worries! We‘ve got your back. Try BYJU‘S free classes today!

t_{1} + 2 t_{2} = 0

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is A $t_{1} t_{2} = - 4$
$A P : A B = 1 : 3$
$∴ A P : P B = 1 : 2$
$∴ P$ has coordinates
$(\frac{a t_{2}^{2} + 2 a t_{1}^{2}}{3}, \frac{2 a t_{2} + 2.2 a t_{1}}{3})$
$P$ also lies on axis of parabola i.e., $y = 0$
$\Rightarrow 2 t_{1} + t_{2} = 0$

Suggest Corrections

Similar questions

Q. Assertion :The locus of the center of the circle described on any focal chord of a parabola

y^{2} = 4 a x

as diameter is

y^{2} = 2 a (x - a)

Reason: If

A (a t_{1}^{2}, 2 a t_{1})

and

B (a t_{2}^{2}, 2 a t_{2})

be the extremities of a focal chord for the parabola

y^{2} = 4 a x,

then

t_{1} t_{2} = - 1

The chord AB of the parabola $y^{2} = 4 a x$ cuts the axis of the parabola at C. If $A = (a t_{1}^{2}, 2 a t_{1}), B = (a t_{2}^{2}, 2 a t_{2})$ and AC : AB = 1:3 then

Q. If a circle intersects the parabola

y^{2} = 4 a x

at point

A (a t_{1}^{2}, 2 a t_{1}), B (a t_{2}^{2}, 2 a t_{2}), C (a t_{3}^{2}, 2 a t_{3}), D (a t_{4}^{2}, 2 a t_{4})

, then

t_{1} + t_{2} + t_{3} + t_{4} + t_{4}

The chord AB of the parabola $y^{2} = 4 a x$ cuts the axis of the parabola at C. If $A = (a t_{1}^{2}, 2 a t_{1}), B = (a t_{2}^{2}, 2 a t_{2})$ and AC : AB = 1:3 then

Q. Match the column
The parabola

y^{2} = 4 a x

has a chord

A B

joining points

A (a t_{1}^{2}, 2 a t_{1})

and

B (a t_{2}^{2}, 2 a t_{2})

Column - I	Column - II
(A) $A B$ is a normal chord if	(P) $t_{2} = - t_{1} - \frac{1}{2}$
(B) $A B$ is a focal chord	(Q) $t_{2} = \frac{- 4}{t_{2}}$
(C) $A B$ subtends $90^{\circ}$ at point $(0, 0)$ if	(R) $t_{2} = \frac{- 1}{t_{1}}$ (S) $t_{2} = - t_{1} - \frac{2}{t_{1}}$

Join BYJU'S Learning Program

Any chord AB of y2=4ax cuts the axis of the parabola at P, so that AP:AB=1:3 where A(at21,2at1),B(at22,2at2), then

The correct option is A t1t2=−4AP:AB=1:3 ∴AP:PB=1:2 ∴P has coordinates (at22+2at213,2at2+2.2at13) P also lies on axis of parabola i.e., y=0 ⇒2t1+t2=0

Any chord $A B$ of $y^{2} = 4 a x$ cuts the axis of the parabola at $P$ , so that $A P : A B = 1 : 3$ where $A (a t_{1}^{2}, 2 a t_{1}), B (a t_{2}^{2}, 2 a t_{2})$ , then

The correct option is A $t_{1} t_{2} = - 4$
$A P : A B = 1 : 3$
$∴ A P : P B = 1 : 2$
$∴ P$ has coordinates
$(\frac{a t_{2}^{2} + 2 a t_{1}^{2}}{3}, \frac{2 a t_{2} + 2.2 a t_{1}}{3})$
$P$ also lies on axis of parabola i.e., $y = 0$
$\Rightarrow 2 t_{1} + t_{2} = 0$