1'6 The Intersection of Straight Line and Conic sections

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1.6 The Intersection of Straight Line and Conic
sections
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1.6.1 To find the points of intersection between
circle and line

Example 1
Find the intersections between the circle x2 y2
24 and line y 2x 5
------ 1
Solution

• Given

------ 2

• by substituting

• from 2 into 1

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Intersection between circle and line

• Therefore

• when

• so the intersection points are

and
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Intersection between circle and line

Example 2
Find the coordinates of the points of
intersection between the circle x2 y2 6x 4y
9 0 and the line y 7 x.
Solution
---------1

• Given

---------2

• by substituting y 7 x into x2 y2 6x 4y
  9 0 gives
• on simplify

• i.e

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Intersection between circle and line

• Given x 5 or x 3

• when

?
?

• so the intersection point, are

and
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1.6.2 To find the points of intersection between
parabola and line
Example 1
The line passing through the point P(p2,2p) on
the curve y2 4x and the point Q(2,0) intersects
the curve once again at the point R. Find the
coordinates of the point R in terms of p.
Solution

• Let the coordinates of R be (r2,2p).

• The gradient of PQ gradient of RQ

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Intersection between parabola and line

• since

• Therefore

?

• Thus

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1.6.3 Intersection between ellipse and line
Example 1
Prove line lx my n 0 touches the ellipse
b2x2 a2y2 a2b2 then a2l2 b2m2 n2
Solution

substituting (1) into (2)
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Intersection between ellipse and line
by using
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1.6.4 Intersection between hyperbola and line
Example 1
Find the condition that the line y mx c shall
touch the hyperbola
Solution

substituting (2) into (1)
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Intersection between hyperbola and line
by using