Tangents are drawn from the point T (2,-1) to the parabola x^2=4y. P and Q are the points of contact of the tangentsa. show that the x-coordinates of P and Q are the roots of the quadratic...

Tangents are drawn from the point T (2,-1) to the parabola x^2=4y. P and Q are the points of contact of the tangents

a. show that the x-coordinates of P and Q are the roots of the quadratic x^2-4x-4=0

b. find the sum of the roots of the equation in part a

c. hence find the midpoint M of the chord PQ, and show that TM is parallel to the axis of the parabola.

 

chord PQ's equation is y=x+1

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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Notice that the problem provides the information that the equation of the chord PQ is `y = x+1` .

The chord PQ intersects the parabola at P and Q, hence, you may find the coordinates of P and Q, solving the following system of equations such that:

`{(y = x+1),(x^2=4y):}`

You need to substitute `x+1`  for `y`  in the second equation such that:

`x^2 = 4(x+1) => x^2 - 4x - 4 = 0`

Notice that this equation coincides with the given equation `x^2-4x-4=0` , hence, the x solutions of this system are the same x solutions of the equation `x^2-4x-4=0` .

b. You should remember the Vieta's relations such that:

`x_1 + x_2 = -b/a`  (sum of roots)

`x_1*x_2 = c/a`  (product of roots)

a,b,c are coefficients of quadratic equation `ax^2 + bx + c = 0`

`x_1 + x_2 = -(-4)/1 => x_1 + x_2 = 4`

Hence, evaluating the sum of roots of equation `x^2-4x-4=0`  yields `x_1 + x_2 = 4` .

c. You need to find the coordinates of the midpoint of chord PQ such that:

`x_M = (x_P+x_Q)/2; y_M = (y_P+y_Q)/2`

You need to find `x_P`  and `x_Q`  solving the equation  x`^2 - 4x - 4 = 0`  such that:

`x_(P,Q) = (4+-sqrt(16+16))/2 => x_(P,Q) = (4+-sqrt32)/2`

`x_(P,Q) = (4+-4sqrt2)/2 => x_(P,Q) = (2+-2sqrt2)`

You need to evaluate `y_P`  and `y_Q`  such that:

`y_(P,Q) = x_(P,Q) + 1 => y_(P,Q) = 3+-2sqrt2`

You may evaluate the coordinates of midpoint M such that:

`x_M = (2+2sqrt2+2-2sqrt2)/2= 2`

`y_M = (3+2sqrt2+3-2sqrt2)/2 = 3`

You need to find the axis of symmetry of parabola, hence, you should remember that the vertical line `x = -b/2a ` represents this axis of symmetry.

The problem provides the equation of parabola `x^2 = 4y` , hence, you need to convert this form in standard form such that:

`y = x^2/4 => x = -0/2(1/4) = 0`

Hence, the y axis represents the axis of symmetry of parabola `x^2 = 4y` .

You need to verify if the line TM ist parallel to y axis, hence, you need to verify if the slope of TM is equal to `tan 90^o`  such that:

`m_(TM) = (y_M - y_T)/(x_M - x_T)`

`m_(TM) = (3+1)/(2-2) = 4/0 = tan 90^o`

Hence, the line TM is parallel to axis of symmetry of the given parabola.

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