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Hyperbola

Section: Mathematics

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Question : 8 of 34

Marks: +1, -0

If the equation of the tangent of the hyperbola

5x^2-9y^2-20x-18y-34=0

which makes an angle

45^{\circ}

with the positive X -axis in positive direction is

x+by+c=0

b^2+c^2=

[AP EAPCET 22 May 2025 S2]

2 or 13
5 or 26
2 or 26
26 or 28

Solution:

The slope of tangent

m=\tan 45^{\circ}=1

Now,

5x^2-9y^2-20x-18y-34=10

\;\Rightarrow 5(x^2-4x)-9(y^2+2y)=34

\;\Rightarrow 5(x^2-4x+4)-9(y^2+2y+1)=34+20-9

\;\Rightarrow 5(x-2)^2-9(y+1)^2=45

\;\Rightarrow \;\frac{(x-2)^2}{9}-\;\frac{(y+1)^2}{5}=1

The equation of the tangent with slope

m

is

y+1=m(x-2) \pm \sqrt{9m^2-5}

Since,

m=1

\;\Rightarrow\;\; y+1=(x-2) \pm \sqrt{9-5}

\;\Rightarrow\;\; y=x-1\;\text{ or }\; y=x-5

⇒ The equation

x-y-1=0

or

x-y-5=0

Comparing with

x+by+c=0

\;\Rightarrow b=-1\;\text{ and }\; c=-1,-5

\;\Rightarrow b^2+c^2=(-1)^2+(-1)^2=1+1=2

\;\;\text{ and }\; b^2+c^2=(-1)^2+(-5)^2=1+25=26

\;\Rightarrow b^2+c^2=2\;\text{ or }\; 26

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