﻿ A variable straight line of slope 4 intersects the hyperbola xy = 1 at two points. The locus of the point which divides the line segment between these two points in the ratio 1 : 2 is : Kaysons Education

# A Variable Straight Line Of Slope 4 Intersects The Hyperbola xy = 1 At Two Points. The Locus Of The Point Which Divides The Line Segment Between These Two Points In The Ratio 1 : 2 Is

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## Question

### Solution

Correct option is

16x2 + 10xy + y2 = 2

P(h, k) be any point on the variable line with slope = 4. Hence it equation is y – k m(x – h)                                        …… (1)

m = 4, let this line (1) cut the hyperbola xy = 1            …… (2)

at A(x1y1) and B(x2y2). By assumption, P divides AB in the ratio 1 : 2 then

Thus ⇒ 2x1 + x2 = 3h , 2y1 + y2 = 3k                             …... (3)

By (2), , put this in (1), we get

⇒ 4x2 – (4h – k) x – 1 = 0                                               …… (4)

x1x2 are roots of (4)                          …… (5)

…… (6)

Solving (3) and (4), for x1 and x2, we get

Putting these in (6)

Locus of (h, k) = 16x2 + 10xy – 2 = 0.

#### SIMILAR QUESTIONS

Q1

Find the equations of tangents to the hyperbola x2 – 4y = 36 which are perpendicular to the line x – y + 4 = 0

Q2

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4x2 – 9y2 =36.

Q3

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Q4

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Q5

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Q6

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Q7

Find the range of ‘a’ for which two perpendicular tangents can be drawn to the hyperbola from any point outside the hyperbola

.

Q8

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Q9

The locus of a variable point whose distance from (–2, 0) is  times its distance from the line , is

Q10

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