1.12: Tangents and normals to a quadratic curve

4PM1/2R/November/2020 — Question 6 · 15 marks

1.12 diagram 1
The curve C\displaystyle C with equation
y=x2−5x+4y=x^2-5x+4
crosses the x\displaystyle x-axis at the points A\displaystyle A and B\displaystyle B, as shown in Figure 3.
(a) Find the coordinates of A\displaystyle A and B\displaystyle B.
(3)
The tangent to C\displaystyle C at A\displaystyle A meets the tangent to C\displaystyle C at B\displaystyle B at T\displaystyle T.
(b) Find the coordinates of T\displaystyle T.
(6)
The normal to C\displaystyle C at A\displaystyle A meets the normal to C\displaystyle C at B\displaystyle B at N\displaystyle N.
(c) Find the coordinates of N\displaystyle N.
(3)
(d) Find the area of quadrilateral ATBN\displaystyle ATBN.
(3)