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Higher Maths Help!

I am currently studying the applications unit in higher maths and I am on the circle topic. I am stuck on this question: Determine whether the line with equation 3y-5x=11 is a chord, tangent, or does not touch the circle with equation x2+y2-14x-8y+31=0. I am not sure how to substitute the equation into the circle equation as if you rearrange it, it comes out as fractions if I am right. Can anyone help me with this? Thank you in advance!
Original post by sabrinaclark23
I am currently studying the applications unit in higher maths and I am on the circle topic. I am stuck on this question: Determine whether the line with equation 3y-5x=11 is a chord, tangent, or does not touch the circle with equation x2+y2-14x-8y+31=0. I am not sure how to substitute the equation into the circle equation as if you rearrange it, it comes out as fractions if I am right. Can anyone help me with this? Thank you in advance!


Yes, rearrainging the linear equation for x or y will give fractions.

Leave the coefficients as fractions and substitute into the equation of the circle.

Do you know how to tell whether it is a chord, tangent or neither?
Original post by Muttley79
Yes, rearrainging the linear equation for x or y will give fractions.

Leave the coefficients as fractions and substitute into the equation of the circle.

Do you know how to tell whether it is a chord, tangent or neither?


Thank you! Yes I did that but I struggled to subsitute the fractions in the equation and multiply out the brackets then collect like terms. I will try again haha, yeah I think so!
Original post by sabrinaclark23
Thank you! Yes I did that but I struggled to subsitute the fractions in the equation and multiply out the brackets then collect like terms. I will try again haha, yeah I think so!
i

I have managed to figure it out however, I am unsure of what I would have to do for this question: The circle with equation x2+y2-10x-14y+29=0 has two tangents with gradient 2. Find the points of contact of the circle and these tangents and hence find the equations of both tangents.

Thank you.
Reply 4
Original post by sabrinaclark23
i

I have managed to figure it out however, I am unsure of what I would have to do for this question: The circle with equation x2+y2-10x-14y+29=0 has two tangents with gradient 2. Find the points of contact of the circle and these tangents and hence find the equations of both tangents.

Thank you.


Sub in y = 2x + c into the equation of the circle and consider discriminants.
Original post by sabrinaclark23
i

I have managed to figure it out however, I am unsure of what I would have to do for this question: The circle with equation x2+y2-10x-14y+29=0 has two tangents with gradient 2. Find the points of contact of the circle and these tangents and hence find the equations of both tangents.

Thank you.


y=2x+cy=2x+c would make the line tangent to the circle at two points, so sub it in and look for when the discriminant of your quadratic is equal to 0 in terms of c, then you can solve for c.

Geometrically it would look like this, where obviously both lines have the same gradient but different constants:
circle.PNG
Original post by Zacken
Sub in y = 2x + c into the equation of the circle and consider discriminants.

Thank you so much, I will try that!
Original post by RDKGames
y=2x+cy=2x+c would make the line tangent to the circle at two points, so sub it in and look for when the discriminant of your quadratic is equal to 0 in terms of c, then you can solve for c.


Geometrically it would look like this, where obviously both lines have the same gradient but different constants:
circle.PNG

Thank you so much, I will try that!
Original post by RDKGames
y=2x+cy=2x+c would make the line tangent to the circle at two points, so sub it in and look for when the discriminant of your quadratic is equal to 0 in terms of c, then you can solve for c.

Geometrically it would look like this, where obviously both lines have the same gradient but different constants:
circle.PNG

I substituted in the equation, multiplied out the brackets and collected like terms however I got
5x2+4cx+c2-38x-14c+29=0 and I don't know how to form a quadratic equation with that, can you help? Thanks
Reply 9
Original post by sabrinaclark23
I substituted in the equation, multiplied out the brackets and collected like terms however I got
5x2+4cx+c2-38x-14c+29=0 and I don't know how to form a quadratic equation with that, can you help? Thanks


Put the coefficients of each term in brackets and you get this 5x2+(4c38)x+(2914c+c2)=0 5x^2+(4c-38)x+(29-14c+c^2)=0 . Assuming you have expanded all correctly.
EDIT. You have expanded incorrectly. The constant term should be 914c+c2 9-14c+c^2 .
(edited 7 years ago)

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