Equation of a tangent to the hyperbola 5x2 - y2 = 5 and which passes through an external point (2, 8) is

Question

Equation of a tangent to the hyperbola 5x2 - y2 = 5 and which passes through an external point (2, 8) is (A) 3x - y + 2 = 0 (B) 3x + y - 14 = 0 (C) 23x - 3y - 22 = 0 (D) 3x - 23y + 178 = 0

Tardigrade Admin · Accepted Answer

Let the tangent be y = mx ± √m2-5 Since it passes through (2, 8) ⇒ (8-2m)2 = m2-5 ⇒ 3m2 - 32m + 69 = 0 ⇒ 3m2 - 9m - 23m + 69 = 0 ⇒ (3m - 23) (m - 3) = 0 ⇒ m = 3 or (23/3)

Q. Equation of a tangent to the hyperbola $5 x^{2} - y^{2} = 5$ and which passes through an external point (2, 8) is

3x - y + 2 = 0

3x + y - 14 = 0

23x - 3y - 22 = 0

3x - 23y + 178 = 0

Let the tangent be $y = m x \pm m^{2} - 5$
Since it passes through (2, 8) $\Rightarrow (8 - 2 m)^{2} = m^{2} - 5$
$\Rightarrow 3 m^{2} - 32 m + 69 = 0 \Rightarrow 3 m^{2} - 9 m - 23 m + 69 = 0 \Rightarrow (3 m - 23) (m - 3) = 0 \Rightarrow m = 3$ or $\frac{23}{3}$

Q. Equation of a tangent to the hyperbola 5x2−y2=5 and which passes through an external point (2, 8) is