The Circle: Equations, Lines and Tangents

Name: The Circle: Equations, Lines and Tangents
Author: StemPy Lab

Where is the centre of a round pond when you know three points on its shore?

Start from the distance of a point from a fixed centre and write the equation of a circle in its standard and general forms.
Then place a point and a line against a circle, find the length of a chord, and draw the tangent and the normal.
You end with the tangents from a point outside, the length of a tangent and the chord of contact.
It covers the first part of the Indian syllabus unit (NCERT, JEE Main) and the matching topic of A-Level Pure Maths.

3 lessons, about 1 h 25 min3 simulations10 problems3 practice questionsPython in every lesson

Course outline (3 lessons)

Lesson 1about 30 min
The equation of a circle
A circle is the set of points at a fixed distance from a fixed point. That one sentence gives the equation $(x - h)^2 + (y - k)^2 = r^2$ . Expanded, it is the general form $x^2 + y^2 + 2gx + 2fy + c = 0$ , with centre $(-g, -f)$ and radius $\sqrt{g^2 + f^2 - c}$ . You can find a circle from three points or from the ends of a diameter.
1 simulation1 practice question1 Python program5 problems1 Python lab
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Lesson 2about 30 min
A line and a circle
Put a point or a line next to a circle and ask where it is. A point is inside, on or outside according to the sign of $S_1$ . A line cuts, touches or misses the circle according to its distance $d$ from the centre. The chord has length $2\sqrt{r^2 - d^2}$ , and the tangent at a point is found by the rule $T = 0$ .
1 simulation1 practice question1 Python program3 problems
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Lesson 3about 25 min
Tangents from an outside point
From a point outside a circle you can draw two tangents, and they have the same length, $\sqrt{S_1}$ . The line through the two points of contact is the chord of contact, with the equation $T = 0$ . This lesson also finds the area of the triangle formed, the pair of tangents, and the director circle, the place from which the tangents are at right angles.
1 simulation1 practice question1 Python program2 problems1 Python lab1 question set1 note
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Board exam format15 questions · 20 min

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