The curve y=x^2-3x-3 crosses the x-axis at P and Q. The tangents to the curve at P and Q meet at R. The normals to the curve at P and Q meet at S.  Find the distance RS.

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You need to remember that x intersections P and Q  represent the roots of the equation `x^2-3x-3 = 0` , hence, you need to use the quadratic formula to find x coordinates of the points P and Q such that:

`x_(P,Q) = (3+-sqrt(9+12))/2 => x_(P,Q) = (3+-sqrt21)/2`

You need to find the equations of the tangent lines to the given curve, at the points P and Q such that:

`y - y_P = f'(x_P)(x - x_P)`

`y - y_Q = f'(x_Q)(x - x_Q)`

You need to evaluate `f'(x_P) ` and`f'(x_Q)`  such that:

`f'(x_P) = (x^2_P-3x_P-3) => f'(x_P) = 2x_P - 3`

Substituting `(3+sqrt21)/2`  for `x_P`  yields:

`f'(x_P) = 2(3+sqrt21)/2 - 3 => f'(x_P) = 3+sqrt21-3`

`f'(x_P) = sqrt21`

You may write the equation of tangent line at the point P such that:

`y - 0 = sqrt21(x - (3+sqrt21)/2) => y = sqrt21(x - (3+sqrt21)/2)`

You need to evaluate `f'(x_Q)`  such that:

`f'(x_Q) = 2x_Q - 3 => f'(x_Q) = -sqrt21`

You may write the equation of tangent line at the point Q such that:

`y - 0 = -sqrt21(x - (3-sqrt21)/2) => y = -sqrt21(x - (3-sqrt21)/2)`

The problem provides the information that the tangent lines intersect at R, hence, you need to solve the following system of equations to find R such that:

`{(y = sqrt21(x - (3+sqrt21)/2)),(y = -sqrt21(x - (3-sqrt21)/2)):}` =>`sqrt21(x - (3+sqrt21)/2) = -sqrt21(x - (3-sqrt21)/2) => x - (3+sqrt21)/2 = (3-sqrt21)/2 - x`

You need to move the terms that contain x to the left side such that:

`x + x = (3-sqrt21)/2+(3+sqrt21)/2 => 2x = 6/2 => 2x = 3 => x = 3/2`

`y =sqrt21(3/2 - (3+sqrt21)/2) => y = sqrt21*(-sqrt21)/2 = -21/2`

Hence, evaluating the coordinates of the point of intersection of tangents at P and Q, yields `R(3/2,-21/2).`

You need to find the equations of the normal lines at P and Q, hence, you need to remember that these normal lines are perpendicular to the tangent lines at P and Q.

You need to remember the equation that relates two perpendicular lines such that:

`m_1*m_P = -1`

`m_1`  represents the slope of tangent line

`m_P ` represents the slope of normal line

Notice that the slope of tangent line at P is `m_1 = sqrt21` , hence, you may evaluate `m_P`  such that:

`m_P = -1/sqrt21`

You need to use the point slope form of equation of line to write the equation of normal line at the point P, such that:

`y - y_P = m_P(x - x_P) => y = (-1/sqrt21)(x - (3+sqrt21)/2)`

Notice that the slope of tangent line at Q is `m_2 = -sqrt21` , hence, you may evaluate `m_Q`  such that:

`m_Q = 1/sqrt21`

You need to use the point slope form of equation of line to write the equation of normal line at the point Q, such that:

`y - y_Q = m_Q(x - x_Q) => y = (1/sqrt21)(x - (3-sqrt21)/2)`

The problem provides the information that the tangent lines intersect at S, hence, you need to solve the following system of equations to find R such that:

`{(y= (-1/sqrt21)(x - (3+sqrt21)/2)),(y = (1/sqrt21)(x - (3-sqrt21)/2)):}` `=> (-1/sqrt21)(x - (3+sqrt21)/2)= (1/sqrt21)(x - (3-sqrt21)/2)` => `-x+ (3+sqrt21)/2 = x - (3-sqrt21)/2` => `2x = 6/2 => 2x = 3 => x = 3/2`

`y = (1/sqrt21)(3/2 - 3/2 + sqrt21/2) => y = 1/2`

Hence, evaluating the point of intersection of normal lines yields `S(3/2,1/2).`

You need to use the distance formula to evaluate the distance RS such that:

`RS = sqrt((x_S - x_R)^2 + (y_S - y_R)^2)`

`RS = sqrt((3/2 - 3/2)^2 + (1/2+ 21/2)^2) `

`RS = sqrt((22/2)^2) => RS = sqrt(11^2) = 11`

Hence, evaluating the distance RS, under the given conditions, yields `RS = 11` .

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