If 2 x - y + 1 = 0 is a tangent to the hyperbola x 2 a 2 - y 2 16 = 1 , then which of the following cannot be sides of a right angled triangle? [2017]

Question

If $2 x - y + 1 = 0$ is a tangent to the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{16} = 1,$ then which of the following cannot be sides of a right angled triangle? [2017]

Smart Achievers · Accepted Answer

(1, 2, 3)

Given $2 x - y + 1 = 0$ i.e. $y = 2 x + 1$ is a tangent to hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{16} = 1$

$∴$ $c^{2} = a^{2} m^{2} - b^{2}$

$\Rightarrow 1^{2} = a^{2} \times 2^{2} - 16$

$\Rightarrow a^{2} = \frac{17}{4}$ $\Rightarrow a = \frac{\sqrt{17}}{2}$

$∴$ $a, 4, 1; a, 4, 2; 2 a, 8, 1$ i.e. $\frac{\sqrt{17}}{2}, 4, 1; \frac{\sqrt{17}}{2}, 4, 2; \sqrt{17}, 8, 1$ cannot be the sides of a right triangle.

Solution

Q.
If $2 x - y + 1 = 0$ is a tangent to the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{16} = 1,$ then which of the following cannot be sides of a right angled triangle [2017]

1 $a, 4, 1$

2 $a, 4, 2$

3 $2 a, 8, 1$

4 $2 a, 4, 1$

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Solution

Q. If 2x-y+1=0 is a tangent to the hyperbola x2a2-y216=1, then which of the following cannot be sides of a right angled triangle [2017]

1 a, 4, 1

2 a, 4, 2

3 2a, 8, 1

4 2a, 4, 1