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

Byju's Answer

Standard XII

Mathematics

Tangent To a Parabola

If the chord ...

Question

If the chord joining the points $t_{1}$ and $t_{2}$ to the parabola $y^{2} = 4 a x$ is normal to the parabola at $t_{1}$ then prove that $t_{1} (t_{1} + t_{2}) = - 2$ .

Open in App

Solution

$y^{2} = 4 a x$
$2 y \frac{d y}{d x} = 4 a$
$y \frac{d y}{d x} = 4 a$
$\frac{d y}{d x} = \frac{4 a}{y}$
Slope of tangent at $t_{1} = \frac{2 a}{2 a t_{1}} = \frac{1}{t_{2}}$
$∴$ Slope of normal at $t_{1} = - t_{1}$
Slope of line joining $t_{1}$ & $t_{2}$
$= (\frac{2 a t_{2} - 2 a t_{1}}{a t_{2}^{2} - a t_{1}^{2}})$
$= \frac{2}{t_{1} + t_{2}}$
Both slopes are same.
So,
$\frac{2}{(t_{1} + t_{2})} = - t_{1}$
$∴ t_{1} (t_{1} + t_{2}) = - 2$ .

Suggest Corrections

Similar questions

t_{1}

and

t_{2}

are two points on the parabola

y^{2} = 4 x

. If the chord joining them is a normal to the parabola at

t_{1}

then

Q. If

t_{1}

and

t_{2}

are two points on the parabola

y^{2} = 4 a x

. If the focal chord joining them coincides with the normal chord, then

Q. The normal at

t_{1}

on the parabola

y^{2} = 4 a x

meets the parabola at

t_{2}

then

(t_{1} + \frac{2}{t_{1}})

is:

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) $t_{2} = - t_{1} - \frac{2}{t_{1}}$

If the chord joining the points $(a t_{1}^{2}, 2 a t_{1})$ and $(a t_{2}^{2}, 2 a t_{2})$ of the parabola $y^{2} = 4 a x$ passes through the focus of the parabola, then

Join BYJU'S Learning Program

If the chord joining the points t1 and t2 to the parabola y2=4ax is normal to the parabola at t1 then prove that t1(t1+t2)=−2.

y2=4ax2ydydx=4aydydx=4adydx=4aySlope of tangent at t1=2a2at1=1t2∴ Slope of normal at t1=−t1Slope of line joining t1 & t2=(2at2−2at1at22−at21)=2t1+t2Both slopes are same.So,2(t1+t2)=−t1∴t1(t1+t2)=−2.

If the chord joining the points $t_{1}$ and $t_{2}$ to the parabola $y^{2} = 4 a x$ is normal to the parabola at $t_{1}$ then prove that $t_{1} (t_{1} + t_{2}) = - 2$ .