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# Higher Maths Help!

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1. 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!
2. (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?
3. (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!
4. (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.
5. (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.
6. (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.
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:
7. (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!
8. (Original post by RDKGames)
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:
Thank you so much, I will try that!
9. (Original post by RDKGames)
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:
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
10. (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 . Assuming you have expanded all correctly.
EDIT. You have expanded incorrectly. The constant term should be .

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