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
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.
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).
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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