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

Two tangents to a parabola meet at an angle

45 °;

prove that the locus of their point of intersection is the curve

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

. If they meet at an angle

60 °

; prove that the locus is,

y^{2} - 3 x^{2} - 10 a x - 3 a^{2} = 0

.

Accepted Answer

(A) We know that the equation of tangent to parabola $y^{2} = 4 a x$ is given by $y = m x + \frac{a}{m} . . . (1)$ .

Let $P (h, k)$ be the external point. So, $P (h, k)$ satisfies equation (1).

Therefore, we have:

$k = m h + \frac{a}{m}$ or $m^{2} h - m k + a = 0 . . . (2)$

Let $m_{1}$ and $m_{2}$ be the slopes of two tangents through $P$ ,
As the angle between them is $45 °$ we get:
$\tan 45^{o} = |\frac{m_{1} - m_{2}}{1 + m_{1} m_{2}}| = |\frac{\sqrt{\{{(m_{1} + m_{2})}^{2} - 4 m_{1} m_{2}\}}}{1 + m_{1} m_{2}}|$

Since, equation (2) is quadratic in terms of $m$ , so we have:

$m_{1} + m_{2} = \frac{k}{h} . . . (3)$ and $m_{1} m_{2} = \frac{a}{h} . . . (4)$ .
Putting the values form equations (3) and (4) we get:
$1 = |\frac{\sqrt{(\frac{k^{2}}{h^{2}} - \frac{4 a}{h})}}{(1 + \frac{a}{h})}|$
Or $1 = |\frac{\sqrt{(k^{2} - 4 a h)}}{h + a}|$
Generalizing after squaring and simplifying, we get the locus of $(h, k)$ as:
$y^{2} - 4 a x = {(x + a)}^{2}$
Proved.

(B) Again, if the angle between the tangents if $60^{o}$ , then
$\tan 60^{o} = |\frac{m_{1} - m_{2}}{1 + m_{1} m_{2}}| = |\frac{\sqrt{\{{(m_{1} + m_{2})}^{2} - 4 m_{1} m_{2}\}}}{1 + m_{1} m_{2}}|$
Or $\sqrt{3} = |\frac{\sqrt{(k^{2} - 4 a h)}}{a + h}|$ [as in part (a)]
Squaring, simplifying and generalizing, the required locus is:
$y^{2} - 3 x^{2} - 10 a x - 3 a^{2} = 0 .$

Proved.

Two tangents to a parabola meet at an angle $45 °;$ prove that the locus of their point of intersection is the curve $y^{2} - 4 a x = {(x + a)}^{2}$ . If they meet at an angle $60 °$ ; prove that the locus is, $y^{2} - 3 x^{2} - 10 a x - 3 a^{2} = 0$ .

Important Questions on The Parabola (Continued)

Learn and Prepare for any exam you want !

Two tangents to a parabola meet at an angle 45°; prove that the locus of their point of intersection is the curve y2-4ax=x+a2. If they meet at an angle 60°; prove that the locus is, y2-3x2-10ax-3a2=0.

Important Questions on The Parabola (Continued)

Learn and Prepare for any exam you want !

Two tangents to a parabola meet at an angle $45 °;$ prove that the locus of their point of intersection is the curve $y^{2} - 4 a x = {(x + a)}^{2}$ . If they meet at an angle $60 °$ ; prove that the locus is, $y^{2} - 3 x^{2} - 10 a x - 3 a^{2} = 0$ .