The equation of one of the tangents drawn from the poin $(0,1)$ to the hyperbola $45 x^2-4 y^2=5$ is

The equation of one of the tangents drawn from the poin $(0,1)$ to the hyperbola $45 x^2-4 y^2=5$ is
  1. $4 y+5=0$
  2. $3 x+4 y-4=0$
  3. $5 x-6 y+6=0$
  4. $9 x-2 y+2=0$

Solution

Let equation of tangent passing $(0,1)$ and slope $m$ be $y-1=m(x-0) \Rightarrow y=m x+1$
Intersection with hyperbola $45 x^2-4 y^2=5$ $\begin{aligned} & 45 x^2-4(m x+1)^2=5 \\ & \Rightarrow\left(45-4 m^2\right) x^2-8 m x-9=0 \end{aligned}$
For $y=m x+1$ to be tangent, discriminant $=0$ $\begin{aligned} & 64 m^2+4\left(45-4 m^2\right) 9=0 \\ & \Rightarrow 16 m^2=405 \Rightarrow m= \pm \frac{9}{2} \end{aligned} $ $\therefore$ Equation tangents are $9 x-2 y+2=0 \text { and } 9 x+2 y-2=0$

Asked in: AP EAMCET 2024 (21 May Shift 1)

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