Question

Find the area of a triangle formed by the line X cos alpha + Y sin alpha = p with the coordinates axes

Answer 1

area of triangle is P²/sin2α

x cosα + ysinα = P is a straight line where P is perpendicular drawn from origin to line and α is angle made by P with x - axis as shown in figure.

here it is clear that, P is altitude of triangle then line joining of two intersecting points is base.

x intercept , y = 0

so, x = P/cosα

and y - intercept , x = 0

y = P/sinα

now length of base = √{P²/cos²α+ P²/sin²α} =P√{1/cos²α + 1/sin²α}

= P√{(sin²α + cos²α)/sin²α.cos²α}

= P√{4/4sin²α.cos²α}

= P/√{4/(2sinα.cosα)²}

= P√{4/sin²2α}

= 2P/sin2α

now area of triangle = 1/2 × P × 2P/sin2α

= P²/sin2α

Answer 2

Answer:

hope you find the attachment