For the curve

MATHEMATICS · Subjective · A Level · Mathematics · Coordinate geometry in the (x y) plane · Coordinate geometry in the (x y) plane

Problem

For the curve

x^{2}+k y^{2}=5k

, where

k

is a non-zero real constant, a tangent is drawn at a point

P(x_{1},y_{1})

lying in the first quadrant. (i) Obtain the expression for

\dfrac{dy}{dx}

at an arbitrary point

(x,y)

on the curve. (ii) Show that the coordinates of

P

satisfy

x_{1}^{2}+k y_{1}^{2}=5k

. (iii) The tangent at

P

is required to pass through the fixed point

(4,0)

. Hence form an equation in

k

and solve for the admissible value(s) of

k

. (iv) Using the value(s) of

k

found in part (iii), write the complete equation of the required tangent. (v) Determine whether the normal at

P

is parallel to the line

2x+y-7=0

, giving reasons for your answer.