Answer:

you answer tell in BRAINLY app

Answer:

Step-by-step explanation:

substitute the value of y in the equation of the curve:

[tex]x^{2} + 2x(2x + k) + 20 = 0\\x^{2} + 4x^{2} + 2kx + 20 = 0\\5x^{2} + 2kx + 20 = 0[/tex]

As the line is tangent to the curve, so the discriminant  [tex]b^{2} - 4ac = 0[/tex]

a = 5,  b = 2k,  c = 20

[tex](2k)^{2} - 4(5)(20) = 0\\4k^{2} - 400 = 0 \\k^{2} - 100 = 0\\k^{2} = 100\\k = +10 \\k = -10[/tex]

Therefore, the equation of the tangent would either be y = 2x + 10  or  

y = 2x - 10

Put k = 10 in the simplified quadratic equation:

[tex]5x^{2} + 2(10)x +20 = 0\\5x^{2} + 20x +20 = 0\\x^{2} + 4x + 4 = 0\\(x+2)^{2} = 0\\x + 2 = 0\\x = -2[/tex]

Put x = -2 in the equation of the tangent y = 2x + 10 to find y:

y = 2(-2) + 10

y = -4 + 10

y = 6

Therefore the point of contact of the curve and the straight line when k = 10 would be (-2, 6)

Now, put k = -10 in the simplified quadratic equation:

[tex]5x^{2} + 2(-10)x +20 = 0\\5x^{2} - 20x +20 = 0\\x^{2} - 4x + 4 = 0\\(x-2)^{2} = 0\\x - 2 = 0\\x = 2[/tex]

Put x = 2 in the equation of the tangent y = 2x - 10 to find y:

y = 2(2) - 10

y = 4 - 10

y = -6

Therefore the point of contact of the curve and the straight line when k = -10 would be (2, -6)