You're viewing 1 free preview question for Maths AI HL — subscribe to unlock 11 more, plus the Full Question Bank, Mock Papers & Worksheets.Upgrade

Coordinate Geometry · Paper 3 — Free Maths AI HL Practice Questions

1ChallengeLAQEquation of a straight line and slope-intercept form10 marksPaper 3~15 min

A telecommunications company is planning to install a fibre-optic cable connecting two towns,

A

and

B

. The cable must pass through a relay station at a point

P

on a straight road running between two substations. Town

A

is at

(2, 5)

and town

B

is at

(10, 11)

, with distances in kilometres. The road passes through substation

X

(0, 2)

and substation

Y

(8, 6)

(a)

Show that the equation of the road

XY

y = \dfrac{1}{2}x + 2

. [3 marks]

(b)

The cable consists of two straight segments, from

A

P

and from

P

B

, where

P

has

x

-coordinate

x

. Show that the total cable length

L

, in kilometres, satisfies

L = \sqrt{(x-2)^2 + \left(\tfrac{1}{2}x - 3\right)^2} + \sqrt{(x-10)^2 + \left(\tfrac{1}{2}x - 5\right)^2}.

[3 marks]

(c)

Using your GDC, find the minimum value of

L

and the coordinates of

P

that achieve this minimum. [2 marks]

(d)

The reflection of

A(2,5)

across the line

y = \dfrac{1}{2}x + 2

is the point

A'(6, 1)

. Explain, using a geometric argument, why the minimum total cable length equals the straight-line distance

A'B

, and verify this numerically. [2 marks]

Solutions

Questions 2-12

Unlock these questions with Premium

That's every free question here — 1 of 12.

+11 more questions, plus Mock Papers & Worksheets, are waiting in Premium.