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Position of two points with respect to a given line

Let the line be ax + by + c = 0 and P(x1, y1), Q(x2, y2) be two points.

Case 1:

If P(x1, y1) and Q(x2, y2) are on the opposite sides of the line 
ax + by + c = 0, then the point R on the line ax + by + c = 0 divides the line PQ internally in the ratio m1 : m2, where m1/m2 must be positive.

Co-ordinates of R

        are (m1x2+m2x1/m1+m2, m1y2+m2y1/m1+m2).

Point R lies on the line ax + by + c = 0.

         m1/m2 = ax1+by1+c/ax2+by2+c > 0

1458_Position of two points with respect to a given line.JPG

So that ax1 + by1 + c and ax2 + by2 + c should have opposite signs.

Case 2:

If ax1 + by1 + c and ax2 + by2 + c have the same signs then m1/m2 = –ve, so that the point R on the line ax + by + c = 0 will divide the line PQ externally in the ratio m1 : m2 and the points P(x1, y1) and Q(x2, y2) are on the same side of the line ax + by + c = 0.

 

Illustration:

Find the range of θ in the interval (0, π) such that the points (3, 5) and (sinθ, cosθ) lie on the same side of the line x + y – 1 = 0.

Solution:

        3 + 5 – 1 =7 > 0  sinθ + cosθ – 1 > 0

         sin(π/4 + θ) > 1/√2

         π/4 < π/4 + θ < 3π/4

         0 < θ < π/2.

 

Illustration:

        Find a, if (α, α2) lies inside the triangle having sides along the lines

                2x + 3y = 1, x + 2y – 3 = 0, 6y = 5x – 1.

Solution:

        Let A, B, C be vertices of the triangle.

        A  (–7, 5), B  (5/4, 7/8),

        C  (1/3, 1/9).

        Sign of A w.r.t. BC is –ve.

1566_triangle having sides along the lines.JPG

If p lies in-side the ¦ABC, then sign of P will be the same as sign of a w.r.t. the line BC

     5α – 6α2 – 1 < 0.                         …… (1)

        Similarly    2α + 3α2 – 1 > 0.                                …… (2)

        And,          α + 2α2 – 3 < 0.                          …… (3)

        Solving, (1), (2) and (3) for α and then taking intersection,

        We get      α ? (1/2, 1) ∪ (–3/2, –1).

 

Illustration:

The equations of the perpendicular bisectors of the sides AB and AC of a triangle ABC are respectively x – y + 5 = 0 and x + 2y = 0. If the co-ordinates of A are (1, –2), find the equation of BC.

Solution:

        From the figure,

        E  (x1+1/2, y1–2/2),

        F  (x2+1/2, y2–2/2).

59_equations of the perpendicular bisectors of sides of triangle.JPG

Alt text : equations of the perpendicular bisectors of sides of triangle

Since E and F lie on OE and OF respectively,

                x1 – y1 + 13 = 0                                  … (1)

        and   x2 + 2y2 – 3 = 0                                  … (2)

        Also, slope of AB = –1 and slope of AC is 2, so that

                x1 + y1 + 1 = 0.                                  … (3)

        And   2x2 – y2 – 4 = 0                                  … (4)

        Solving these equations, we get the co-ordinates of B and C as

                B  (–7, 6) and C ≡ (11/5, 2/5)

 Equation of BC is 14x + 23y – 40 = 0.

 

Illustration:

Two fixed points A and B are taken on the co-ordinate axes such that OA = a and OB = b. Two variable points A’ and B’ are taken on the same axes such that OA’ + OB’ = OA + OB. Find the locus of the point of intersection of AB’ and A’B.

Solution:

        Let A ≡ (a, 0), B (0, b), A’  (a’, 0), B’  (0, b’).

        Equation of A’B is x/a' + y/b' = 1.                                          …. (1)

        and equation of AB’ is x/a + y/b'  = 1.                                    …. (2)

        Subtracting (1) from (2), we get, x (1/a – 1/a') + y(1/b' – 1/b) = 0.

         x(a'–a)/aa' + y(b–b')/bb' = 0.                             [Using a’ – a = b – b’]

         x/a(b–b'+a) + y/bb', 0  b’ = a(a+b)y/ay–bx.                              ….. (3)

        From (2) b’x + ay = (4) we get x + y = a + b

        which is the required locus.

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