Parallel and perpendicular lines

The circle C has centre A with coordinates (7, 5).

The line l, with equation y 2x 1, is the tangent to C at the point P, as shown in Figure 3.

Show that an equation of the line PA is 2y x 17

Find an equation for C.

The line with equation y 2x + k, k 1 is also a tangent to C.

Find the value of the constant k.

A curve with equation y = f(x) passes through the point (4, 9). Given that

find f(x), giving each term in its simplest form.

Point P lies on the curve . The normal to the curve at P is parallel to the line 2 y + x = 0

Find the x coordinate of P.

Figure 2 shows a right angled triangle LMN. The points L and M have coordinates (–1, 2) and (7, –4) respectively.

Find an equation for the straight line passing through the points L and M. Give your answer in the form ax + by + c = 0, where a, b and c are integers.

Given that the coordinates of point N are (16, p), where p is a constant, and angle LMN = 90°,

find the value of p.

Given that there is a point K such that the points L, M, N, and K form a rectangle,

find the y coordinate of K.

Find the gradient of L,. (2) The line L, is perpendicular to L, and passes through the point (2, 5).

Figure 2 shows a sketch of the curve H with equation y = + 4, x

Give the coordinates of the point where H crosses the x-axis.

Give the equations of the asymptotes to H.

Find an equation for the normal to H at the point P (–3, 3).

This normal crosses the x-axis at A and the y-axis at B.

Find the length of the line segment AB. Give your answer as a surd.

The line l₁ has equation y = 2x + 3 The line l₂ is perpendicular to l₁ and passes through the point (5, 6).

Find an equation for l₂ in the form ax + by + c = 0, where a, b and c are integers.

The line l₂ crosses the x-axis at the point A and the y-axis at the point B.

Find the x-coordinate of A and the y-coordinate of B.

Given that O is the origin,

find the area of the triangle OAB.

The curve C has equation

Find , giving each term in its simplest form. The point P on C has x-coordinate equal to

Find the equation of the tangent to C at the point P, giving your answer in the form y = a x + b, where a and b are constants.

The tangent to C at the point Q is parallel to the line with equation 2x − 3y + 18 = 0

Find the coordinates of Q.

The line L₁ has equation 4y + 3 = 2x The point A (p, 4) lies on L₁

Find the value of the constant p.

The line L₂ passes through the point C (2, 4) and is perpendicular to L₁

Find an equation for L₂ giving your answer in the form ax + by + c = 0, where a, b and c are integers.

The line L₁ and the line L₂ intersect at the point D.

Find the coordinates of the point D.

Show that the length of CD is 5 A point B lies on L₁ and the length of AB = The point E lies on L₂ such that the length of the line CDE = 3 times the length of CD.

Find the area of the quadrilateral ACBE.

The line l₁ has equation 3x 5y 2 0

Find the gradient of l₁.

The line l₂ is perpendicular to l₁ and passes through the point (3, 1).

Find the equation of l₂ in the form y = mx + c, where m and c are constants.

The points A and B have coordinates (6, 7) and (8, 2) respectively. The line l passes through the point A and is perpendicular to the line AB, as shown in Figure 1.

Find an equation for l in the form ax + by + c = 0, where a, b and c are integers.

Given that l intersects the y-axis at the point C, find

the coordinates of C,

the area of OCB, where O is the origin.

The curve C has equation

The point P has coordinates (2, 7).

Show that P lies on C.

Find the equation of the tangent to C at P, giving your answer in the form y = mx + c, where m and c are constants.

The point Q also lies on C. Given that the tangent to C at Q is perpendicular to the tangent to C at P,

show that the x-coordinate of Q is 2 6.

The curve C has equation

The point P on C has x-coordinate equal to 2.

Show that the equation of the tangent to C at the point P is y = 1 – 2 x.

Find an equation of the normal to C at the point P.

The tangent at P meets the x-axis at A and the normal at P meets the x-axis at B.

Find the area of triangle APB.

The curve C has equation y = kx³ – x² + x – 5, where k is a constant.

The point A with x-coordinate – lies on C. The tangent to C at A is parallel to the line with equation 2 y – 7 x + 1 = 0. Find

the value of k,

the value of the y-coordinate of A.

The points Q (1, 3) and R (7, 0) lie on the line l₁, as shown in Figure 2. The length of QR is a5.

Find the value of a.

The line l₂ is perpendicular to l₁, passes through Q and crosses the y-axis at the point P, as shown in Figure 2. Find

the coordinates of P,