Parallel & perpendicular lines

The circle x² + y² + 2x - 14y + 25 = 0 has its centre at the point C. The line 7y = x + 25 intersects the circle at points A and B.

Prove that triangle ABC is a right-angled triangle.

Quadrilateral KLMN has vertices K (- 4, 1), L (5, - 1), M (6, 2) and N (2, 5), as shown in Fig. 6.1.

Find the coordinates of the following midpoints.

• P, the midpoint of KL

• Q, the midpoint of LM

• R, the midpoint of MN

• S, the midpoint of NK

Verify that PQRS is a parallelogram.

Show that = k(-a + b - c), where k is a constant to be determined.

Verify that AH is parallel to DE.

Verify that BC is parallel to GF.

In this question you must show detailed reasoning. Fig. 10 shows the curve given parametrically by the equations x = y = - for t > 0.

Find the coordinates of the point on the curve at which the tangent to the curve is parallel to

the line 4 y + x = 1.

Find the cartesian equation of the curve. Give your answer in factorised form.

Show that the line which passes through the points (2, - 4) and (- 1, 5) does not intersect the line

3 x + y = 10.

In this question you must show detailed reasoning.

The curve xy + y² = 8 is shown in Fig. 9.

Find the coordinates of the points on the curve at which the normal has gradient 2.

Fig. 12 shows the circle (x-1)² + (y+1)² = 25, the line 4y = 3x-32 and the tangent to the circle at the point A (5, 2). D is the point of intersection of the line 4 y = 3 x - 32 and the tangent at A.

Write down the coordinates of C, the centre of the circle.

(A) Show that the line 4 y = 3 x - 32 is a tangent to the circle.

(B) Find the coordinates of B, the point where the line 4 y = 3 x - 32 touches the circle.

Prove that ADBC is a square.

The point E is the lowest point on the circle. Find the area of the sector ECB.