The equation of a curve is y^2+2x = 13 and the equation of a line is 2y+x = k, where k is a constant. (ii) Find the value of k for which the line is a tangent to the curve.

aruv | Student

Given equation of the curve is


`` which can be rewritten as

`y^2=-2x+13`           (i) , which is an equation of parabola.

A straight line intersect parabola at most two points.


2y+x=K            (ii) ,

can intersect  (i)  at most two points. (i) intersect (ii) if



`x^2+(8-2K)x+K^2-52=0`  (iii)

has real roots i.e x has real values. Since  (ii) is tangents to (i),therefore roots of the equation (ii) are equal.

roots will equal if discriminant of quadratic equation (iii) in x is zero.







Thus K=8.5 then line 2y+x=k 2 will tangent to the given curve.

Black line K>8.5

green line k=8.5

blue line  k<8.5

See green is tangent to the curve.


llltkl | Student

First, the slope of the line `2y+x = k` has to be found.

`2y+x = k`

`rArr y=1/2(k-x)` (where k is a constant)

`rArr y'=-1/2`

slope = -1/2

So we have to find where curve `y^2+2x=13` has slope = -1/2


`rArr 2y*y'=-2`

`rArr y'=-1/y`

Put the condition for tangency,


`rArr y=2`

Put this value of y in the equation of the curve,


`rArr x=9/2`

Now put the values of x and y to get k as:


`rArr k=17/2`

Therefore, for `k=17/2` , the line `2y+x=k` is tangent to the curve `y^2+2x=13` , at the point (`9/2, 2` ).