Equations of Lines

A line $L$ has intercepts $a$ and $b$ on the coordinate axes. When the axes are rotated through a given angle, keeping the origin fixed, the same line has intercepts $p$ and $q$. Which of the following is true?

The straight lines $\dfrac{x}{a} + \dfrac{y}{b} = k$ and $\dfrac{x}{a} + \dfrac{y}{b} = \dfrac{1}{k}$ (with $k\neq0$) meet on:

The equation of the base of an equilateral triangle is x + y = 2 and the vertex is (2, –1). The length of the side of the triangle is.

The lines 3x – 4y + 4 = 0 and 6x – 8y – 7 = 0 are tangent to the same circle. The radius of the this circle is.

The equations of the line parallel to the line $2x – 3y = 7$ and passing through the middle point of the line segment joining the points (1, 3) and (1, –7) is.

If the straight line ax + by + c = 0 always passes through (1, –2), then a, b, c are in

If $(– 4, 5)$ is one vertex and $7x – y + 8 = 0$ is one diagonal of a square, then the equation of the other diagonal is

If (2, 1), (–1, –2), (3, 3) are the midpoints of the sides BC, CA, AB of a triangle ABC, then equation of the line BC is

The foot of the perpendicular from the point (2, 4) upon $x + y = 1$ is

The equation of the tangent line to the curve y = 2x sin x at the point (π/2, π), is

If the lines x + (a – 1)y + 1 = 0 and 2x + a²y – 1 = 0 are perpendicular, then the condition satisfies by a is