Straight Lines
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If the line, 2x – y + 3 = 0 is at a distance 1/√5 and 2/√5 from the lines 4x – 2y + α = 0 and 6x – 3y + β = 0, respectively, then the sum of all possible values of α and β is
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If the line, 2x – y + 3 = 0 is at a distance 1/√5 and 2/√5 from the lines 4x – 2y + α = 0 and 6x – 3y + β = 0, respectively, then the sum of all possible values of α and β is
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2x-y+3=0
4x-2y+α=0 ⇒ 2x-y+α/2=0
6x-3y+β=0 ⇒ 2x-y+β/3=0
d? = |α/2-3|/√ (2²+1²) = 1/√5 ⇒ |α-6|=2 ⇒ α-6=2, -2 ⇒ α=8,4
d? = |β/3-3|/√ (2²+1²) = 2/√5 ⇒ |β-9|=6 ⇒ β-9=6, -6 ⇒ β=15,3
Sum of all value of α and β = 30.
Upvote 
Let PQ be a diameter of the circle x² + y² = 9. If α and β are the lengths of the perpendiculars from P and Q on the straight line, x + y = 2 respectively, then the maximum value of αβ is.
Let PQ be a diameter of the circle x² + y² = 9. If α and β are the lengths of the perpendiculars from P and Q on the straight line, x + y = 2 respectively, then the maximum value of αβ is.
Let P (2cosθ, 2sinθ)
∴ Q (-2cosθ, -2sinθ)
Given line x+y-2=0
∴ α = |2cosθ + 2sinθ – 2| / √2
β = |-2cosθ - 2sinθ – 2| / √2
∴ αβ = √2 (cosθ + sinθ – 1) · √2 (cosθ + sinθ + 1)
= 2|cos²θ + sin²θ + 2sinθcosθ – 1| = 2|sin2θ|
Max |sin2θ| = 1
∴ maximum αβ = 2.
Upvote If the perpendicular bisector of the line segment joining the points P(1,4) and Q(k, 3) has y-intercept equal to -4, then a value of k is:
If the perpendicular bisector of the line segment joining the points P(1,4) and Q(k, 3) has y-intercept equal to -4, then a value of k is:
Any point (x, y) on perpendicular bisector equidistant from p and q
∴ (x − 1)² + (y − 4)² = (x − k)² + (y − 3)²
At x = 0, y = -4
∴ 1 + 64 = k² + 49
k² = 16
Upvote 