Hello guys!
I've been trying to figure out how to get the correct answer in these problems. The Cambridge books only state the answers but not how to get them. There are the following:
1. The line y=mx+c is tangent to y^2 + (x-10)^2 = 25. Find m.
(This is the simplified form. I keep getting 1/3, but apparently it's incorrect. Help would be really appreciated!)
There is not enough information to give a specific numerical answer to this question - are you given a point through which the tangent passes?
2. Let P(x1,y1) be a point on the circle x^2 + y^2 = a^2. Find the gradient of the line which is tangent to the circle at P.
A tangent to a circle at a point is always perpendicular to the diameter at that point. The diameter of the circle x^2 + y^2 = a^2, passing through P(x1, y1) on the circle, has equation y – y1 = (y1/x1)(x – x1). Hence the tangent passing through P(x1, y1) is y – y1 = –(x1/y1)(x – x1).
You can also set up a quadratic equation for the point of intersection, and then set the discriminant equal to zero to find the gradient.
3. An equilateral triangle ABC circumscribes x^2 + y^2 = a^2. The side BC has the equation x=-a. Find the equations of AB and AC, and hence the equation of the circle.
Each angle in an equilateral triangle is 60°. Since the side BC is perpendicular to the x-axis, the other sides must form angles of 150° from the positive x-axis and -150° from the positive x-axis, respectively. Then use m = tan(θ) to find the gradients of the sides AB and AC. You can also use trigonometry to find the coordinates of A, B, C, and then the equations of the lines follow.
(Is there any topic I need to revise? I am not too familiar with this, or I may have forgotten.)
Knowing some circle geometry is often useful when dealing with questions about intersecting lines and circles.
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