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What is the point of contact when the tangent line y equals x plus c touches the circle x squared plus y squared equals 4?
The line meets the circle when:
y = x + c
→ x² + y² = 4
→ x² + (x + c)² - 4 = 0
→ x² + x² +2cx + c² - 4 = 0
→ 2x² + 2cx + (c² - 4) = 0
If the line is a tangent to the circle, this has a repeated root. This occurs when the discriminant is 0, ie when:
(2c)² - 4 × 2 × (c² - 4) = 0
→ 4c² - 8c² + 32 = 0
→ 4c² = 32
→ c² = 8
→ c = √8 = 2√2
This can now be substituted back into the meeting equation above:
2x² + 2cx + (c² - 4) = 0
→ 2x² + 2 × 2√2 × x + (√8)² - 4 = 0
→ 2x² + 4√2 x + 8 - 4 = 0
→ x² + 2√2 x + 2 = 0
→ (x + √2)² = 0
→ x = -√2
→ y = x + 2√2
= -√2 + 2√2
= √2
→ point of contact is (-√2, √2)
What else can I help you with?
Tangent of a circle is what?
A tangent of a circle is a straight line that touches the circle at only one point.
How do you find the tangent of a circle?
a tangent is a line that touches the circle at only ONE point
A line that touches a circle?
Tangent
Definition of tangent in a circle?
A tangent to a circle is a line from a point outside the circles which touches the circle at a single point.
How can you recognize a tangent of a circle?
A tangent of a circle is a straight line that touches the circle's circumference at just one point.
A line that touches the circle at one point?
A tangent is a line that touches a circle at exactly one point. It is perpendicular to the radius at the point of contact.
Tangent of a circle is what?
A tangent of a circle is a straight line that touches the circle at only one point.
How do you find the tangent of a circle?
a tangent is a line that touches the circle at only ONE point
A line that touches a circle?
Tangent
What is the tangent of a circle?
A striaght line , outside the circle, that just touches the circle in one place.
Definition of tangent in a circle?
A tangent to a circle is a line from a point outside the circles which touches the circle at a single point.
How can you recognize a tangent of a circle?
A tangent of a circle is a straight line that touches the circle's circumference at just one point.
What does '' a tangent to a circle '' mean?
It is a straight line that touches the circle.
What is the straight line which touches the circumference in a circle?
It is the tangent of the circle
The point where the tangent line intersects a circle?
A tangent line touches the circumference of a circle at just one point but it does not intersect the circle.
What is the equation of the tangent line that touches the circle x squared plus y squared -8x -16y -209 equals 0 at a coordinate of 21 and 8?
Circle equation: x^2 +y^2 -8x -16y -209 = 0 Completing the squares: (x-4)^2 +(y-8)^2 = 289 Centre of circle: (4, 8) Radius of circle 17 Slope of radius: 0 Perpendicular tangent slope: 0 Tangent point of contact: (21, 8) Tangent equation: x = 21 passing through (21, 0)
A line that intersects a circle at exactly one point?
Such a line is called a tangent line or a tangent to the circle. [Tangent is Latin for touching-- a tangent line touches the circle at just one point. ]
