Name: Pair of Tangent, Chord of Contact
Uploaded: 2019-07-19
Duration: 19 min 37 s
Description: This video contains all the details of the chord of the contact pair of tangent and cord bisected at a point. This concept is very useful for the students.

Question

Show that the locus of the point of intersection of two tangents with which the tangent at the vertex form triangle of constant area

c^{2}

is the curve

x^{2} (y^{2} - 4 a x) = 4 c^{4} .

Accepted Answer

Let the equation of the parabola is $x^{2} (y^{2} - 4 a x) = 4 c^{4} .$ , then the equation of the tangents at the points $P (a {t_{1}}^{2}, 2 a t_{1})$ and $Q (a {t_{2}}^{2}, 2 a t_{2})$ are

$y t_{1} = x + a {t_{1}}^{2} . . . (2)$ and $y t_{2} = x + a {t_{2}}^{2} . . . (3)$ respectively,

Both the tangent intersect at $T (a t_{1} t_{2,}, a (t_{1} + t_{2,}))$ and with the tangents at the vertices $R (0, a t_{1})$ and $S (0, a t_{2})$ as shown in the figure. Now we have to find the locus of the point $T$ under the condition that the area of the triangle $△ T R S$ is $c^{2}$ .

So taking $T (h, k)$

$∴ h = a t_{1} t_{2} . . . (4)$ and $k = a (t_{1} + t_{2}) . . . (5)$

From the figure area of $a r (△ T R S) = \frac{1}{2} |a (t_{1} - t_{2})| |a t_{1} t_{2}|$

$a r (△ T R S) = \frac{1}{2} a |(t_{1} - t_{2})| |h|$ (from the equation $(4)$ )

$\Rightarrow a r (△ T R S) = \frac{1}{2} a |h| \sqrt{{(t_{1} + t_{2})}^{2} - 4 t_{1} t_{2}}$

$\Rightarrow a r (△ T R S) = \frac{1}{2} a |h| \sqrt{{(\frac{k}{a})}^{2} - 4 (\frac{h}{a})}$ (from equations $(4)$ and $(5)$

$\Rightarrow c^{2} = \frac{1}{2} a |h| \sqrt{{(\frac{k}{a})}^{2} - 4 (\frac{h}{a})}$ (Area of triangle is given)

$\Rightarrow {(2 c^{2})}^{2} = a^{2} {|h|}^{2} ({(\frac{k}{a})}^{2} - 4 (\frac{h}{a}))$

$\Rightarrow 4 c^{4} = a^{2} h^{2} (\frac{k^{2} - 4 a h}{a^{2}})$

$\Rightarrow 4 c^{4} = h^{2} (k^{2} - 4 a h)$

Generalising, we get the locus as $x^{2} (y^{2} - 4 a x) = 4 c^{4}$ . Hence proved.

Show that the locus of the point of intersection of two tangents with which the tangent at the vertex form triangle of constant area $c^{2}$ is the curve $x^{2} (y^{2} - 4 a x) = 4 c^{4} .$

Important Questions on The Parabola (Continued)

Learn and Prepare for any exam you want !

Show that the locus of the point of intersection of two tangents with which the tangent at the vertex form triangle of constant area c2 is the curve x2y2-4ax=4c4.

Important Questions on The Parabola (Continued)

Learn and Prepare for any exam you want !

Show that the locus of the point of intersection of two tangents with which the tangent at the vertex form triangle of constant area $c^{2}$ is the curve $x^{2} (y^{2} - 4 a x) = 4 c^{4} .$