![]() C intersects the X-axis at the points A and B. You can be asked to find the coordinates of the points of intersection of a circle with the x-axis.Įxample: C is a circle with centre (0,3) and radius of 5Īnswer: Label (0,3) with h and k. (You could also draw a sketch of the circle on the x and y axis, plot the given point and the centre point, use a ruler to join the point to the centre and continue from the centre to the edge of the circle where you will find (8,-1).) INTERSECTION OF CIRCLES add 5 onto 3 = 8, and ADD the y coordinates i.e. Now ADD the x coordinate of the translation onto the x coordinate of the centre i.e. How far is it from the y coordinate of the point to the y coordinate of the centre i.e. How far is it from the x coordinate of the point to the x coordinate of the centre i.e. What we are doing is finding how far it is from the point to the centre, then going the same distance from the centre to this new point.Įxample: ( -2, 5 ) is a point on the circle (x-3) 2 + (y-2) 2 = 34Īnswer: The centre of the circle is (3 ,2) However, if the centre is not (0,0) and we are asked the same question, then we use a translation to find another point. Putting the point of a compass on the origin (0,0), as this is the centre, stretch the compass to 3 and draw a circle.Įxample: The point (-4,3) is on the circle x 2 + y 2 = 25Īnswer: The centre of this circle is (0,0) ( there are no numbers after the x and y)Īny point with the numbers 4 and 3 in it will be on thecircle. This circle has a radius of 3 ( square root of 9)ĭraw an x and y axis, marking to 3 on each axis, How to draw a given circle:Įxample: Draw the circle with equation x 2 + y 2 = 9 ![]() If your figure on the left is greater than that on the right, then the point would be outside the circle. (-2-3) 2 + (6-4) 2 = 36 (put all figures on left into calculator)Ģ9 < 36 therefore the point is inside the circle. To show whether a point is inside, outside or on a circle: Label the given point and sub into the equation of the given circle.Įxample 1: Show that the point (2,-3) is on the circle x 2 + y 2 = 16Īnswer: Labelling the point x and y we see that x=2 and y=-3ġ6 = 16 therefore (2,-3) is ON the circleĮxample 2: Show that the point (-2, 6) is inside the circle Write down the centre and radius of this circle.Īnswer : The centre is (0,0) (as no numbers after the x and y) and as r 2 = 25 then the radius will be 5 which is the square root of 25.Įxample 4: Given the equation of a circle is (x-5) 2 + (y+3) 2 = 49Īnswer: The centre is (5,-3) (The sign is the opposite to what it was insideĪnd the radius is 7 which is the square root of 49. Our Equation is therefore, (x-3) 2 + (y-4) 2 = 6 2įinish by squaring the 6, so our answer is (x-3) 2 + (y-4) 2 = 36Įxample 3: Given the equation of a circle is x 2 + y 2 = 25 We label the centre (h,k) so h=3 and k=4 and sub into the Formula. This circle has numbers at its centre, therefore we use Formula 2. We simply write Formula 1 and replace r 2 with 4 2Įxample 2: A circle with centre at (3,4) and a radius of 6: write its equation. This circle has centre (0,0) so we use Formula 1. There are 2 types of question you can be asked: one in which you are given the centre and radius and asked to write the equation of the circle and another where you are given the equation and asked to write down the centre and radius of the given equation.Įxample 1: A circle with centre (0,0) and a radius of 4: write its equation. What does it mean then, if your quadratic that you end up with has 2 solutions? or 1 solution? or no solutions?Well, it would mean that the line goes straight through the circle, or just touches it (is a tangent), or doesn't intersect it at all.FORMULA 1 : Used for a circle which has a centre of (0,0) and a given radius.įORMULA 2 : Used for a circle which has a centre of (h,k) (which means it has numbers, not zeros) and a given radius. When you solve these simultaneously algebraically, you are really finding where the graphs intersect!So, if you were to solve them simultaneously you would end up with a quadratic (either in terms of x or y depending on what you substitute) - but hang-on. It is the part under the square root sign, in other words it can't be negative or there are no real solutions to your quadratic equation.We also know that if it equals 0 there is 1 repeated solution, and if it is greater than 0 then there are 2 distinct solutions.So how does this all relate to coordinate geometry?Well, remember how you can talk about solving equations as the intersection of curves/lines? For example solving 2x = 3x+ 4 could be interpreted as finding the intersection of the line y = 2x and the line y = 3x + 4.So now say you had a circle and a line. ![]() Where does the discriminant come from? It is from the quadratic formula.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |