The equation of a circle with centre $$$C$$$ is $$$x^{2}+y^{2}-8 x+4 y-5=0$$$. The point $$$P(1,2)$$$ lies on the circle. Show that the equation of the tangent to the circle at $$$P$$$ is $$$4 y=3 x+5$$$. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mathematics

IGCSE&ALevel

CAIE

Exam No：9709_s20_qp_12 Year：2020 Question No：11(b)

Answer:

$P(1,2)$ to $C(4,-2)$ has gradient $-\frac{4}{3}$
(FT on coordinates of $C$ )
Tangent at $P$ has gradient $=\frac{3}{4}$
Equation is $y-2=\frac{3}{4}(x-1)$ or $4 y=3 x+5$

Knowledge points:

1.3.1 find the equation of a straight line given sufficient information

1.3.2 interpret and use any of the forms in solving problems (Including calculations of distances, gradients, midpoints, points of intersection and use of the relationship between the gradients of parallel and perpendicular lines.)

1.3.4 use algebraic methods to solve problems involving lines and circles (Including use of elementary geometrical properties of circles, e.g. tangent perpendicular to radius, angle in a semicircle, symmetry.) (Implicit differentiation is not included.)

Answer:

Knowledge points:

Solution:

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