If p and q are the lengths of perpendiculars from the origin to the lines x cos θ – y sin θ = k cos 2θ and x sec θ + y cosec θ = k, respectively, prove that p² + 4q² = k².

Question

TextDistance of a Point from a Line Formula

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Find the distance of the point (3, – 5) from the line 3x – 4y –26 = 0.

Find the distance of the point (–1, 1) from the line 12(x + 6) = 5(y – 2).

Find the points on the x-axis, whose distances from the line x/3 + y/4 = 1 are 4 units.

Find the coordinates of the foot of perpendicular from the point (–1, 3) to the line 3x – 4y – 16 = 0.

In the triangle ABC with vertices A (2, 3), B (4, –1) and C (1, 2), find the equation and length of altitude from the vertex A.

If p is the length of perpendicular from the origin to the line whose intercepts on the axes are a and b, then show that 1/p² = 1/a² + 1/b².