a) Find the eccentricity and foci of the hyperbola x^225 y^216 = 1 . [2] b) Find the ratio in which the line joining the points P ( 2, 4, 7) and Q (3, 5, 1) is divided by the ZX plane. [2] c) If a =…

a) Eccentricity and foci of $\frac{x^2}{25}-\frac{y^2}{16}=1$

The standard form of a horizontal hyperbola is

$$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$$

Comparing, we have

$$a^2=25\Rightarrow a=5,b^2=16\Rightarrow b=4$$

For a hyperbola

$$e=\sqrt{1+\frac{b^2}{a^2}}$$

$$e=\sqrt{1+\frac{16}{25}}=\sqrt{\frac{41}{25}}=\frac{\sqrt{41}}{5}$$

The foci lie on the transverse axis at $(\pm c,0)$ where

$$c^2=a^2+b^2=25+16=41\Rightarrow c=\sqrt{41}$$

$$\boxed{\text{Foci }(\pm \sqrt{41},0)}$$

$$\boxed{\text{Eccentricity }e=\frac{\sqrt{41}}{5}}$$

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b) Ratio in which the line $PQ$ is divided by the $ZX$-plane

The $ZX$-plane is the plane $y=0$.
Let the point of intersection be $R(x,0,z)$.

Parametric equations of $PQ$:

$$x=-2+t(3-(-2))=-2+5t$$ $$y=4+t(-5-4)=4-9t$$ $$z=7+t(-1-7)=7-8t$$

Set $y=0$:

$$4-9t=0\Rightarrow t=\frac{4}{9}$$

Coordinates of $R$:

$$x_R=-2+5\left(\frac{4}{9}\right)=-2+\frac{20}{9}=\frac{-18+20}{9}=\frac{2}{9}$$ $$z_R=7-8\left(\frac{4}{9}\right)=7-\frac{32}{9}=\frac{63-32}{9}=\frac{31}{9}$$

Thus $R$ divides $PQ$ internally in the ratio

$$PR:RQ=t:(1-t)=\frac{4}{9}:\frac{5}{9}=4:5$$

$$\boxed{4:5}$$

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c) Magnitude of $2\vec{a}+\vec{b}-\vec{c}$

Vectors:

$$\vec{a}=(3,-1,-4),\vec{b}=(-2,4,-3),\vec{c}=(-5,7,-1)$$

Compute the linear combination:

$$2\vec{a}=(6,-2,-8)$$ $$2\vec{a}+\vec{b}=(6-2,-2+4,-8-3)=(4,2,-11)$$ $$2\vec{a}+\vec{b}-\vec{c}=(4-(-5),2-7,-11-(-1))=(9,-5,-10)$$

Magnitude:

$$\left|2\vec{a}+\vec{b}-\vec{c}\right|=\sqrt{9^2+(-5{)}^2+(-10{)}^2}=\sqrt{81+25+100}=\sqrt{206}$$

$$\boxed{\sqrt{206}}$$