Standard +0.3 This is a standard coordinate geometry question requiring students to find the circle's radius, the line's equation, and verify tangency by showing the perpendicular distance from centre to line equals the radius. While it involves multiple steps (distance formula, line equation, perpendicular distance formula), these are all routine AS-level techniques with no novel insight required, making it slightly easier than average.
In this question you must show detailed reasoning.
The centre of a circle C is at the point \((-1, 3)\) and C passes through the point \((1, -1)\). The straight line L passes through the points \((1, 9)\) and \((4, 3)\). Show that L is a tangent to C. [7]
Question 8:
8 | (x1)2 (y3)2 r2
r2 (11)2 (13)2
L: m = ‒ 2
y = ‒ 2x + 11 oe
substitution of their y = ‒ 2x + 11 in their
(x1)2 (y3)2 20
x2 ‒ 6x + 9 = 0 oe
(x ‒ 3)2 = 0 so repeated root
Hence line touches the curve and is a tangent | M1
M1
B1
B1
M1
A1
E1
[7] | 2.1
1.1
1.1
1.1
1.1
1.1
2.4 | Left side correct and =
Or find L first (B1B1), then find
equation of line perp to L through
(‒1, 3) (M1M1) then substitute
(M1), solve (A1) then check (E1).
soi
or (‒ 6)2 ‒ 4×1×9 = 0 | or line through centre which is
perpendicular to L has
equation
y ‒ 3 = ½(x ‒ ‒ 1)
meets L at (3, 5)
(3 + 1)2 + (5 ‒ 3)2 = r2 =20 so
lines meet at circumference of
circle at
right angles so L is a tangent
In this question you must show detailed reasoning.
The centre of a circle C is at the point $(-1, 3)$ and C passes through the point $(1, -1)$. The straight line L passes through the points $(1, 9)$ and $(4, 3)$. Show that L is a tangent to C. [7]
\hfill \mbox{\textit{OCR MEI AS Paper 2 2018 Q8 [7]}}