Angle between Two Lines

Q. Question 6 (i)
Name the type of quadrilateral formed, if any, by the following points, and give reasons for your answer:
(i) (- 1, - 2), (1, 0), (- 1, 2), (- 3, 0)

Q. Question 6 (ii)
Name the type of quadrilateral formed, if any, by the following points, and give reasons for your answer:
(ii) (- 3, 5), (3, 1), (0, 3), (- 1, - 4)

Q. Find the distance between the points P(3, 4) and Q(6, 9) using Pythagoras theorem.

Q. The equation of a straight line which cuts off an intercept of $5$ units on negative direction of y-axis and makes an angle ${120}^{\circ }$ with positive direction of x-axis is

1. $y+\sqrt{3}x+5=0$
2. $y-\sqrt{3}x+5=0$
3. $y+\sqrt{3}x-5=0$
4. $y-\sqrt{3}x-5=0$

Q. The triangle formed by $x+3y=1$ and $9x^2-12xy+k{y}^2=0$ is right angled triangle and $k\ne -9,$ then $k=$

1. $7$
2. $5$
3. $1$
4. $3$

Q. Let the vectors $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ and $\overrightarrow{d}$ be such that $(\overrightarrow{a}\times \overrightarrow{b})\times (\overrightarrow{c}\times \overrightarrow{d})=0$. Let $P_1$and $P_2$be planes determined by pair of vectors $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c},\overrightarrow{d}$ respectively. Then the angle between $P_1$and $P_2$ is

1. $0^{\circ }$
2. $\frac{\pi }{2}$
3. $\frac{\pi }{4}$
4. $\frac{\pi }{3}$

Q. Graphically, find the number of solution for the following pair of linear equations in two variables:
6x – 3y + 10 = 0
2x – y + 9 = 0

Q. The equation of the internal bisector of angle BAC of the triangle ABC whose vertices are $A(5,2),B(2,3),C(6,5)$ is

1. $6x+y-32=0$
2. $x-4y+3=0$
3. $2x+y-12=0$
4. $y-2=0$

Q. Question 6
The points A(4, 3), B(6, 4), C(5, -6) and D(-3, 5) are vertices of a parallelogram.

Q. A triangle is formed by the lines whose combined equation is given by $(x+y-4)(xy-2x-y+2)=0$. The equation of its circumcircle is

1. $x^2+y^2-5x-3y-8=0$
2. $x^2+y^2-5x-3y+8=0$
3. $x^2+y^2-3x-5y+8=0$
4. none of these

Q. If a chord of the circle $x^2+y^2-4x-2y-c=0$ is trisected at the points $(\frac{1}{3},\frac{1}{3})$ and $(\frac{8}{3},\frac{8}{3}),$ then

1. Length of the chord $=7\sqrt{2}$
2. $c=20$
3. radius of the circle $=25$
4. $c=25$

Q. For what values of m does the system of equations $3x+my=m$ and $2x-5y=20$ has a solution satisfying the condition x > 0, y > 0.
The ans is $m\in (-\infty -\frac{15}{2})\cup (k,\infty )$
Find the value of k?

Q. Name the type of quadrilateral formed, if any, by the following points, and give reasons for your answer:
(- 1, - 2), (1, 0), (- 1, 2), (- 3, 0)

Q. Equations of diagonals of square formed by lines $x=0,y=0,x=1$ and $y=1$ are

1. $y=x,y+x=1$
2. $y=x,x+y=2$
3. $2y=x,y+x=\frac{1}{3}$
4. $y=2x,y+2x=1$

Q. Are the given points collinear?
The points are (1, – 1), (5, 2) and (9, 5).

1. Yes
2. No
3. Maybe

Q.

Name the type of quadrilateral formed, if any, by the following points, and give reasons for your answer:
(- 1, - 2), (1, 0), (- 1, 2), (- 3, 0)

Q. The base of equilateral triangle with side 2 lies along the y-axis such that the mid-point of the base is at the origin. Find vertices of the triangle.

Q. Find the equation of line which passes through point $(2,3)$ and makes an angle of ${45}^o$ with $x-$axis.

Q. When the axes are rotated through an angle ${45}^{\circ },$ the transformed equation of a curve is $17x^2-16xy+17y^2=225.$ Find the original equation of the curve.

Q. Equation of two equal sides of a triangle are the lines $7x+3y-20=0$ and $3x+7y-20=0$ and the third side passes through the point $(-3,3)$ then the equation of the third side can be -

1. $x+y=0$
2. $x-y+4=0$
3. $x+3=0$
4. $y=3$

Q. Find the equation of the line having an inclination ${30}^{\circ }$ with the positive direction of $X$axis and cut off an intercept of $6$ on the positive side of $Y$ axis.