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    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!
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    (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?
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    (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!
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    (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.
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    (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.
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    (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+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:
    Name:  circle.PNG
Views: 97
Size:  8.7 KB
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    (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!
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    (Original post by RDKGames)
    y=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:
    Name:  circle.PNG
Views: 97
Size:  8.7 KB
    Thank you so much, I will try that!
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    (Original post by RDKGames)
    y=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:
    Name:  circle.PNG
Views: 97
Size:  8.7 KB
    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
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    (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  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  9-14c+c^2 .
 
 
 
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