(x, y) 平面上的坐標幾何：練習題

A straight line \(L_1\) has the equation \(3x + 6y = 5\). Find the gradient of a line \(L_2\) that is perpendicular to \(L_1\).

A circle \(C\) has the equation \(x^2 + y^2 - 10x + 4y - 20 = 0\). Find the coordinates of the centre and the radius of \(C\).

A circle \(C\) has the equation \((x - 4)^2 + (y + 1)^2 = 25\) . A line \(L\) is tangent to \(C\) at the point \(P(7, 3)\) . Find the equation of the line \(L\) .

A circle \(C\) has the equation \((x+1)^2 + (y-3)^2 = 25\). Find the equation of the tangent to the circle at the point \(P(2, 7)\), giving your answer in the form \(ax + by + c = 0\).

Line \(L_1\) has the equation \(2x + 3y = 7\). Line \(L_2\) is perpendicular to \(L_1\) and passes through the point of intersection of the lines \(x - y = 1\) and \(y = 5\). Find the equation of \(L_2\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\), and \(c\) are integers.

The line \( L_1 \) passes through the points \( A(1, 4) \) and \( B(5, 2) \). Find the equation of the line \( L_2 \) which is perpendicular to \( L_1 \) and passes through the point \( P(3, 1) \), giving your answer in the form \( ax + by + c = 0 \).

The points $A$, $B$ and $C$ have coordinates $A(3, 1)$, $B(7, 4)$, and $C(p, 1)$, where $p$ is a constant.

Part (a)

Find the gradient of the line segment $AB$.

Part (b)

The line $L$ passes through $B$ and is perpendicular to $AB$. Find the equation of the line $L$, giving your answer in the form $ax + by + c = 0$, where $a, b,$ and $c$ are integers.

Part (c)

Given that the area of the triangle $ABC$ is 9 square units, find the possible values of $p$.

The circle \(C\) has the equation \(x^2 + y^2 - 4x + 2y - 20 = 0\).
(a) Find the coordinates of the centre \(M\) and the radius of \(C\).
(b) The point \(P(-1, 3)\) lies on the circle. Find the equation of the tangent to the circle at the point \(P\), giving your answer in the form \(ax + by + c = 0\), where \(a, b,\) and \(c\) are integers.
(c) The tangent at \(P\) intersects the \(x\)-axis at the point \(A\) and the \(y\)-axis at the point \(B\). Find the area of the triangle \(OAB\), where \(O\) is the origin.
(d) There is another tangent to the circle \(C\) that is parallel to the tangent found in part (b). Find the equation of this tangent in the form \(ax + by + d = 0\).