Answer to Question #266030 in Differential Equations for Mashi

Question

At a particular point of curve y=2x² -x +q, the equation of tangent is y= 3x-5, Find the.value of the constant q

Answer 1

Solution;

Concept;

Let (a,b) be the point on the curve where the tangent is drawn.

Hence (a,b) is a point on both the curve and the tangent.

If the curve if y=f(x) ,the gradient at that point is;

m=f'(x=a)

And the equation of the tangent is ;

y-b=m(x-a).....(i)

Applying this concept;

Given;

Equation of tangent is ;

$y=3x-5$

Rewrite in the form of (i);

$y=3(x-1)-2$

$y+2=3(x-1)$

This implies that ;

$m=3$

Also given, equation of the curve;

$y=2x^2-x+q$

Differentiate;

$m=f'(x)=4x-1$

At x=1;

$f'(x)=4(1)-1=3=m$

Therefore ,the tangent of the curved is drawn at a point (1,-2)

Substituting into the equation;

$y=2x^2-x+q$

$-2=2(1^2)-1+q$

$q=-3$