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If: y = x +c and y = 3 -x -5x^2

Then: x +c = 3 -x -5x^2

Transposing terms: 5x^2 +2x +(c -3) = 0

Using the discriminant formula: 2^2 -4*5*(c -3) = 0 => c = 16/5

Using the quadratic equation formula: 5x^2 -2x +1/5 = 0 => x = -1/5

By substitution: y = 3

Therefore the point of contact is made at: (-1/5, 3)

Q: What is the point of contact when the tangent line y equals x plus c touches the curve y equals 3 -x -5x squared?

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A tangent is a line that just touches a curve at a single point and its gradient equals the rate of change of the curve at that point.

k = 0.1

In trig, the secant squared divided by the tangent equals the hypotenuse squared divided by the product of the opposite and adjacent sides of the triangle.Details: secant = hypotenuse/adjacent (H/A) and tangent = opposite/adjacent (A/O);Then secant2/tangent = (H2/A2)/(O/A) = H2/A2 x A/O = H2/AO.

They are +/- 5*sqrt(2)

Equation: x² + y² -6x +4y = 0 Completing the squares: (x-3)² + (y+2)² = 13 Centre of circle: (3, -2) Contact point: (6, -4) Slope of radius: -2/3 Slope of tangent: 3/2 Tangent equation: y - -4 = 3/2(x-6) => 2y - -8 = 3x-18 => 2y = 3x-26 Tangent line equation in its general form: 3x-2y-26 = 0

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-2

It is (-0.3, 0.1)

(2, -2)

A tangent is a line that just touches a curve at a single point and its gradient equals the rate of change of the curve at that point.

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)

k = 0.1

In trig, the secant squared divided by the tangent equals the hypotenuse squared divided by the product of the opposite and adjacent sides of the triangle.Details: secant = hypotenuse/adjacent (H/A) and tangent = opposite/adjacent (A/O);Then secant2/tangent = (H2/A2)/(O/A) = H2/A2 x A/O = H2/AO.

-2

It works out that the tangent line of y -3x -5 = 0 makes contact with the circle x^2 +y^2 -2x +4y -5 = 0 at the coordinate of (-2, -1) on the coordinated grid.

They are +/- 5*sqrt(2)

Equation: x² + y² -6x +4y = 0 Completing the squares: (x-3)² + (y+2)² = 13 Centre of circle: (3, -2) Contact point: (6, -4) Slope of radius: -2/3 Slope of tangent: 3/2 Tangent equation: y - -4 = 3/2(x-6) => 2y - -8 = 3x-18 => 2y = 3x-26 Tangent line equation in its general form: 3x-2y-26 = 0

If y = 2x +10 and y^2 = 10x then by forming a single quadratic equation and solving it the point of contact is made at (5/8, 5/2)