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# At how many points does line L intersect with the parabola

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Manager
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At how many points does line L intersect with the parabola  [#permalink]

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01 Feb 2012, 18:16
1
6
00:00

Difficulty:

55% (hard)

Question Stats:

62% (01:55) correct 38% (01:42) wrong based on 316 sessions

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At how many points does line L intersect with the parabola represented by y = x² - 3x + 6?

(1) Line L is parallel to x-axis.
(2) Line L passes through the point (0, 16).
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At how many points does line L intersect with the parabola  [#permalink]

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02 Feb 2012, 08:19
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At how many points does line L intersect with the parabola represented by y = x² - 3x + 6?

Generally a parabola and a line can have 0, 1, or 2 intersection points.

Notice than in our case, since the coefficient of x^2 is positive, then we have an upward parabola, which intersects y-axis at (0, 6). Just to illustrate, below is the graph of y = x^2 - 3x + 6:
Attachment:

parabola.gif [ 3.02 KiB | Viewed 5756 times ]

(1) Line L is parallel to x-axis. All three cases are possible: line L can be below the given parabola (no intersection), line L can go through the vertex (1 intersection) or cross it above the vertex (2 intersections). Not sufficient.

Notice that if it were: line L is parallel to y-axis then we would know that it would cross a parabola at one point, hence in this case the statement would be sufficient.

(2) Line L passes through the point (0, 16). Since (0, 16) is higher than y-intercept of parabola, then line L is "trapped" and must have at least one intersection with parabola (it'll have only one for example if L is x=0). Not sufficient.

(1)+(2) Line L intersect with the parabola at two points. Sufficient.

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01 Feb 2012, 20:44
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At how many points does line L intersect with the parabola represented by y = x² - 3x + 6?

y = x² - 3x + 6?

Since a (coefficient of x²) is +ve, it's an upward parabola.

y intercept is 6

(1) Line L is parallel to x-axis.
Not sufficient, as we don't know where does it intersects the y axis.

(2) Line L passes through the point (0, 16).
Not sufficient as we don't know the slope, depending on slope line may intersect parabola at zero or multiple points.

(1) + (2)

Sufficient, lines intersects parabola at only two points i.e. where y = 16.

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02 Feb 2012, 00:26
a. means y=0,substituting in the equation we get two values for x.
sufficient.

b y= mx + c meaning y = 0+ 16 (substitution)
clearly no clue about other point and slope.
Not sufficient.

A it is.
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Re: At how many points does line L intersect with the parabola  [#permalink]

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02 Feb 2012, 00:31
Smita04 wrote:
At how many points does line L intersect with the parabola represented by y = x² - 3x + 6?
(1) Line L is parallel to x-axis.
(2) Line L passes through the point (0, 16).

What is the source? You did not specify....
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Re: At how many points does line L intersect with the parabola  [#permalink]

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27 Jun 2013, 21:29
Bumping for review and further discussion*. Get a kudos point for an alternative solution!

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Re: At how many points does line L intersect with the parabola  [#permalink]

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23 Aug 2014, 14:52
1
Bunuel wrote:

(2) Line L passes through the point (0, 16). Since (0, 16) is higher than y-intercept of parabola, then line L is "trapped" and must have at least one intersection with parabola (it'll have only one for example if L is y=0). Not sufficient.

.

I think the highlight words should be x = 0, instead of y = 0.
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Re: At how many points does line L intersect with the parabola  [#permalink]

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24 Aug 2014, 12:19
Bunuel wrote:

(2) Line L passes through the point (0, 16). Since (0, 16) is higher than y-intercept of parabola, then line L is "trapped" and must have at least one intersection with parabola (it'll have only one for example if L is y=0). Not sufficient.

.

I think the highlight words should be x = 0, instead of y = 0.

Typo edited. Thank you.
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Re: At how many points does line L intersect with the parabola  [#permalink]

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04 Oct 2014, 00:05
Bunuel wrote:
At how many points does line L intersect with the parabola represented by y = x² - 3x + 6?

Generally a parabola and a line can have 0, 1, or 2 intersection points.

Notice than in our case, since the coefficient of x^2 is positive, then we have an upward parabola, which intersects y-axis at (0, 6). Just to illustrate, below is the graph of y = x^2 - 3x + 6:
Attachment:
parabola.gif

(1) Line L is parallel to x-axis. All three cases are possible: line L can be below the given parabola (no intersection), line L can go through the vertex (1 intersection) or cross it above the vertex (2 intersections). Not sufficient.

Notice that if it were: line L is parallel to y-axis then we would know that it would cross a parabola at one point, hence in this case the statement would be sufficient.

(2) Line L passes through the point (0, 16). Since (0, 16) is higher than y-intercept of parabola, then line L is "trapped" and must have at least one intersection with parabola (it'll have only one for example if L is x=0). Not sufficient.

(1)+(2) Line L intersect with the parabola at two points. Sufficient.

Hi Bunuel ,Can you explain preferably thru diagram how line intersects parabola in 0 and 1 way. I am ok with 2 but not 0 and 1. Please explain.
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Posts: 53066
Re: At how many points does line L intersect with the parabola  [#permalink]

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04 Oct 2014, 02:58
snehamd1309 wrote:
Bunuel wrote:
At how many points does line L intersect with the parabola represented by y = x² - 3x + 6?

Generally a parabola and a line can have 0, 1, or 2 intersection points.

Notice than in our case, since the coefficient of x^2 is positive, then we have an upward parabola, which intersects y-axis at (0, 6). Just to illustrate, below is the graph of y = x^2 - 3x + 6:
Attachment:
The attachment parabola.gif is no longer available

(1) Line L is parallel to x-axis. All three cases are possible: line L can be below the given parabola (no intersection), line L can go through the vertex (1 intersection) or cross it above the vertex (2 intersections). Not sufficient.

Notice that if it were: line L is parallel to y-axis then we would know that it would cross a parabola at one point, hence in this case the statement would be sufficient.

(2) Line L passes through the point (0, 16). Since (0, 16) is higher than y-intercept of parabola, then line L is "trapped" and must have at least one intersection with parabola (it'll have only one for example if L is x=0). Not sufficient.

(1)+(2) Line L intersect with the parabola at two points. Sufficient.

Hi Bunuel ,Can you explain preferably thru diagram how line intersects parabola in 0 and 1 way. I am ok with 2 but not 0 and 1. Please explain.

Check below:
Attachment:

Untitled.png [ 11.72 KiB | Viewed 3258 times ]

Any line which is below the vertex of parabola, for example red line in diagram, won't intersect the parabola.
Any vertical line (for example orange line) or line passing through the vertex (green line) will intersect parabola at one point.

Hope it's clear.
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Re: At how many points does line L intersect with the parabola  [#permalink]

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11 Oct 2018, 05:53
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Re: At how many points does line L intersect with the parabola   [#permalink] 11 Oct 2018, 05:53
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