Rectangular Cartesian Coordinate System

Q.

An equilateral triangle is inscribed in the parabola $y^2=4ax$ so that one angular point of the triangle is at the vertex of the parabola. Find the length of each side of the triangle.

Q. Angle between tangents drawn to $x^2+y^2-2x-4y+1=0$ at the points where it is cut by the line $y=2x+c,is\frac{\pi }{2}$ then

1. 
2. 
3. 
4. 

Q. Write the equations for the x and y -axes.

Q.

Find the length of the perpendicular from the point $(x^1,y^1)$ to the straight line Ax + By + C = 0, the axes being inclined at an angle $\omega$, and the equation being written such that C is a negative quantity.

1. $\frac{A{x}^1+B{y}^1+C}{\sqrt{A^2+B^2}}.cos\omega$

2. $\frac{A{x}^1+B{y}^1+C}{\sqrt{A^2+B^2}}.sin\omega$

3. $\frac{A{x}^1+B{y}^1+C}{\sqrt{A^2+B^2-2ABsin\omega }}.cos\omega$

4. $\frac{A{x}^1+B{y}^1+C}{\sqrt{A^2+B^2-2ABcos\omega }}.sin\omega$

Q.

The straight lines joining origin and the point of intersections of the curve $y^2$ =4x with the line x+y+n = 0 are perpendicular. Find the value of $n^2$.

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Q.

Find the length of the perpendicular from the point $(x^1,y^1)$ to the straight line Ax + By + C = 0, the axes being inclined at an angle $\omega$, and the equation being written such that C is a negative quantity.

1. $\frac{A{x}^1+B{y}^1+C}{\sqrt{A^2+B^2}}.cos\omega$

2. $\frac{A{x}^1+B{y}^1+C}{\sqrt{A^2+B^2}}.sin\omega$

3. $\frac{A{x}^1+B{y}^1+C}{\sqrt{A^2+B^2-2ABsin\omega }}.cos\omega$

4. $\frac{A{x}^1+B{y}^1+C}{\sqrt{A^2+B^2-2ABcos\omega }}.sin\omega$

Q. The number of integral values of m, for which the x-co-ordinate of the point of intersection of the lines 3x + 4y = 9 and y = mx + 1 is also an integer is

1. 2
2. 0
3. 4
4. 1

Q. Let $z_1,z_2,\&z_3$ be the vertices and $z_0$ be the circumcentre of an equilateral triangle. then $z_1^2+z_2^2+z_3^2$ =

1. $z_0^2$
2. $-z_0^2$
3. $3z_0^2$
4. $-3z_0^2$