A level maths pure

A curve has equation x^2 y^2 12x = 64
A line has equation y = mx 10

In the case that the line intersects the curve at two distinct points, show that (20m 12)^2 - 144(m^2 1) > 0

Pleas help me solve this. I have got up to x^2 m^2x^2 20mx 100 12x = 64
Thankyou for any help!

(edited 5 years ago)

Original post by fr51494

A curve has equation x^2 y^2 12x = 64
A line has equation y = mx 10

In the case that the line intersects the curve at two distinct points, show that (20m 12)^2 - 144(m^2 1) > 0

Pleas help me solve this. I have got up to x^2 m^2x^2 20mx 100 12x = 64
Thankyou for any help!

Good.

Now get it into the form

ax^2 + bx + c = 0

by grouping the

x^2, x

and constant terms.

Then start looking at the discriminant.

Reply 2

username3062024

8

To show that there are two points of intersection the discriminant has to be bigger than 0

Reply 3

fr51494

OP

8

Thank you both so much, that makes perfect sense!
So it would become:
(1 + m^2)x^2 + (20m + 12)x + 36 = 0
a = (1 + m^2), b = (20m + 12), c = 36
Using the discriminant, b^2 - 4ac > 0
(20m + 12)^2 - 4 x 46 (m^2 + 1) > 0
= (20m + 12) ^2 - 144 (m^2 + 1) > 0

Reply 4

sugarhigh

1

Original post by fr51494

A curve has equation x^2 y^2 12x = 64
A line has equation y = mx 10

In the case that the line intersects the curve at two distinct points, show that (20m 12)^2 - 144(m^2 1) > 0

Pleas help me solve this. I have got up to x^2 m^2x^2 20mx 100 12x = 64
Thankyou for any help!

how did you get the 20mx. I'm assuming you squared the second equation and then substituted it in

A level maths pure

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