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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 $t_{1} (t_{1} + t_{2}) = - 2$ .

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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,
$\frac{2}{(t_{1} + t_{2})} = - t_{1}$
$∴ t_{1} (t_{1} + t_{2}) = - 2$ .

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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 $t_{1} (t_{1} + t_{2}) = - 2$ .