Consider the parabola y2=20 x, ellipse (x2/16)+(y2/9)=1 and hyperbola (x2/29)-(y2/4)=1 The equation of common tangents to all of them, is

Question

Consider the parabola y2=20 x, ellipse (x2/16)+(y2/9)=1 and hyperbola (x2/29)-(y2/4)=1 The equation of common tangents to all of them, is (A) y= ±(x-5) (B) y= ±(x+3) (C) y= ±(x+5) (D) y= ±(x+4)

Tardigrade Admin · Accepted Answer

If y = mx + c is a common tangent, so c =(5/ m )= ± √16 m 2+9= ± √29 m 2-4 ⇒ m 2=1 ⇒ m = ± 1 So, y= ±(x+5)

Q. Consider the parabola $y^{2} = 20 x$ , ellipse $\frac{x ^{2}}{16} + \frac{y ^{2}}{9} = 1$ and hyperbola $\frac{x ^{2}}{29} - \frac{y ^{2}}{4} = 1$
The equation of common tangents to all of them, is

$y = \pm (x - 5)$

$y = \pm (x + 3)$

$y = \pm (x + 5)$

$y = \pm (x + 4)$

If $y = m x + c$ is a common tangent, so
$c = \frac{5}{m} = \pm 16 m^{2} + 9 = \pm 29 m^{2} - 4 \Rightarrow m^{2} = 1 \Rightarrow m = \pm 1$
So, $y = \pm (x + 5)$

Q. Consider the parabola y2=20x, ellipse 16x2​+9y2​=1 and hyperbola 29x2​−4y2​=1 The equation of common tangents to all of them, is

y=±(x−5)

y=±(x+3)

y=±(x+5)

y=±(x+4)