Parametric Form of a Straight Line

Q. The parallelism condition for two straight lines one of which is specified by the equation ax+by+c=0 the other being represented parametrically by $x=\alpha t+\beta ,y=\gamma t+\delta$ is given by

1. $\alpha \gamma -b\alpha =0,\beta =\delta =c=0$
2. $a\alpha -b\gamma =0,\beta =\delta =0$
3. $a\alpha +b\gamma =0$
4. $a\gamma =b\alpha =0$

Q. Find the equation of the straight lines passing through the point $(-10,4)$ and making an angle $\theta$ with the line x-2y=10 , such that tan $\theta$ =2.

Q.

For all values of $\theta$, the lines represented by the equation
$(2cos\theta +3sin\theta )x+(3cos\theta -5sin\theta )y-(5cos\theta -2sin\theta )=0$

1. pass through a fixed point

2. pass through the point (1, 1)

3. pass through a fixed point whose reflection in the line
   $x+y=\sqrt{2}$ is$(\sqrt{2}-1,\sqrt{2}-1)$

4. pass through the origin

Q. The Bubna dam has four inlets.Through the first three inlets, the dam can be filled in 12 minutes; through the second, the third and the fourth inlet, it can be filled in 15 minutes; and through the first and the first inlet, in 20 minutes.How much time will take all the four inlets to fill up the dam?

1. $8min$
2. $12min$
3. None of these
4. $10min$

Q. Find the equation of the largest circle with centre $(1,0)$ that can be inscribed in the ellipse $x^2+4y^2=16$.

1. $(x-1{)}^2+y^2=\frac{11}{3}$
2. $(x-1{)}^2+(y-1{)}^2=\frac{11}{3}$
3. $(x{)}^2+(y-1{)}^2=\frac{11}{3}$
4. $(x-1{)}^2+y^2=\frac{-11}{3}$