Let A be a point on the x-axis. Common tangents are drawn from A to the curves x2+y2=8 and y2=16 x. If one of these tangents touches the two curves at Q and R, then (Q R)2 is equal to

Question

Let A be a point on the x-axis. Common tangents are drawn from A to the curves x2+y2=8 and y2=16 x. If one of these tangents touches the two curves at Q and R, then (Q R)2 is equal to (A) 76 (B) 81 (C) 72 (D) 64

Tardigrade Admin · Accepted Answer

y = mx +(4/ m ) (|(4/ m )|/√1+ m 2)=2 √2 ∴ m =± 1 y =± x ± 4 Point of contact on parabola Let m =1,(( a / m 2), (2 a / m )) R (4,8) Point of contact on circle Q(-2,2) ∴( QR )2=36+36=72

Q. Let $A$ be a point on the $x$ -axis. Common tangents are drawn from $A$ to the curves $x^{2} + y^{2} = 8$ and $y^{2} = 16 x$ . If one of these tangents touches the two curves at $Q$ and $R$ , then $(QR)^{2}$ is equal to

76

81

72

64

$y = m x + \frac{4}{m}$
$\frac{∣ \frac{4}{m} ∣}{1 + m ^{2}} = 22$
$∴ m = \pm 1$
$y = \pm x \pm 4$ Point of contact on parabola
Let $m = 1, (\frac{a}{m ^{2}}, \frac{2 a}{m})$
$R (4, 8)$
Point of contact on circle $Q (- 2, 2)$
$∴ (QR)^{2} = 36 + 36 = 72$

Q. Let A be a point on the x-axis. Common tangents are drawn from A to the curves x2+y2=8 and y2=16x. If one of these tangents touches the two curves at Q and R, then (QR)2 is equal to

76

81

72

64

y=mx+m4​ 1+m2​∣m4​∣​=22​ ∴m=±1 y=±x±4 Point of contact on parabola Let m=1,(m2a​,m2a​) R(4,8) Point of contact on circle Q(−2,2) ∴(QR)2=36+36=72

Q. Let $A$ be a point on the $x$ -axis. Common tangents are drawn from $A$ to the curves $x^{2} + y^{2} = 8$ and $y^{2} = 16 x$ . If one of these tangents touches the two curves at $Q$ and $R$ , then $(QR)^{2}$ is equal to

$y = m x + \frac{4}{m}$
$\frac{∣ \frac{4}{m} ∣}{1 + m ^{2}} = 22$
$∴ m = \pm 1$
$y = \pm x \pm 4$ Point of contact on parabola
Let $m = 1, (\frac{a}{m ^{2}}, \frac{2 a}{m})$
$R (4, 8)$
Point of contact on circle $Q (- 2, 2)$
$∴ (QR)^{2} = 36 + 36 = 72$