Tangent problem - 10 points?

Substitute (x + c) for y in the second equation.

x² + y² = 4

x² + (x + c)² = 4

2x² + 2cx + (c² - 4) = 0

The solutions to this equation are the x-coordinates of the points of intersection. There can be only one point of intersection, so there is exactly one solution to the quadratic equation. The discriminant is zero.

(2c)² - 4(2)(c² - 4) = 0

c² - 2(c² - 4) = 0

-c² + 8 = 0

c² = 8

c = ±2√(2)

First, find when the slope of the circle is 1 (since the slope of y = x + c is 1)

x^2 + y^2 = 4

2x * dx + 2y * dy = 0

x * dx + y * dy = 0

y * dy = -x * dx

dy / dx = -x / y

dy/dx = 1

-x / y = 1

-x = y * 1

y = -x

x^2 + y^2 = 4

x^2 + (-x)^2 = 4

x^2 + x^2 = 4

2x^2 = 4

x^2 = 2

x = +/- sqrt(2)

y = -x

y = -/+ sqrt(2)

(sqrt(2) , -sqrt(2)) and (-sqrt(2) , sqrt(2)) are the points of tangency

y = x + c

sqrt(2) = -sqrt(2) + c

2sqrt(2) = c

y = x + 2 * sqrt(2)

y = x + c

-sqrt(2) = sqrt(2) + c

-2sqrt(2) = c

y = x - 2 * sqrt(2)

c = +/- 2 * sqrt(2)

Tan 30 is opposite divided by employing adjoining. So in a 30-60-ninety triangle component ratio, the different could be a million (30 is the least attitude, so the smallest component could correspond to it) and the adjoining could be sq. root of three, because of the fact the hypotenuse is two. so a million/sq. root of three is your answer. Tan 40 5 is the comparable element, different than the different is a million and the adjoining is a million, so tan 40 5 is a million/a million= a million. Drawing a image of the triangles is effective too. desire I helped!

the line is ------------- y = x+ c ---------------------------(I)

circle is x^3 +y^2 = 4 -------------------------(I

if the line (i ) is tangent to circle (II) then perpendicular from centre (0,0) of circle

will be equal to RADIUS ( 2 ) of the circle .

so - c/ sqrt( 1^2+1^2) = 2

or c = 2 sqrt2 ANSWER

Note :

Line y = mx + c touches circle x² + y² = a²

iff : c = ± a√(m²+1). ............................................. (1)

Here : a = 2, m = 1.

Hence : c = ± 2√(1²+1) = ± 2√2 ...................................... Ans.

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