If values of m for which the line y=mx+25 touches the hyperbola 16x2-9y2=144 are roots of the equation x2-(a+b)x-4=0 , then value of (a + b) is equal to

A
2
B
4
C
zero
D
5

Solution:

$Given hyperbola 16 x^{2} - 9 y^{2} = 144 \Rightarrow \frac{x^{2}}{9} - \frac{y^{2}}{16} = 1 w h e r e a^{2} = 9, b^{2} = 16 If y = mx + 2 \sqrt{5} is tangent then c^{2} = a^{2} m^{2} - b^{2} \Rightarrow 20 = 9 m^{2} - 16 \Rightarrow 9 m^{2} = 36 \Rightarrow m^{2} = 4 \Rightarrow m = \pm 2 G i v e n e q u a t i o n x^{2} - (a + b) x - 4 = 0 G i v e n - 2, 2 a r e r o o t s o f t h i s e q u a t i o n sum of roots = a + b \Rightarrow - 2 + 2 = a + b \Rightarrow a + b = 0$

If values of m for which the line $y = mx + 2 \sqrt{5}$ touches the hyperbola $16 x^{2} - 9 y^{2} = 144$ are roots of the equation $x^{2} - (a + b) x - 4 = 0$ , then value of (a + b) is equal to

Solution:

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