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Help with these two differentiation questions

1) The curve y = x^2 - 3x -4 crosses the x axis at P and Q. The tangents to the curve P and Q meet at R. The normals to the curve at P and Q meet at S. Find the distance RS.

2) The line y = 6x - 7 is a tangent to the curve y = x^2 + k. Find k.

Thanks

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Reply 1

krisshP

Original post by Alex1397

what did you try?

Reply 2

Kvothe the Arcane

Original post by Alex1397

1) The curve y = x^2 - 3x -4 crosses the x axis at P and Q. The tangents to the curve P and Q meet at R. The normals to the curve at P and Q meet at S. Find the distance RS.
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(i)Think of the y value at the x-axis
(ii)Find P and Q with that information
(iii)Find the gradient of the curve. Substitute the x values of P and Q into
gradient.
(iii)Use the gradient value to find the equations of the tangents using the formula

y-y_1=m(x-x_1)

(iv)Solve the simultaneous equations to find the point of intersection, R.
(v)Substitute the x values of P and Q into the negative inverse of the gradient (as it's a normal: recall

m_1m_2=-1

)
(vi)Use the formula

y-y_1=m(x-x_1)

and the new gradient to find the equations of the normals.
(vii)Solve the simultaneous equations to find the point of intersection, S.
(viii)Apply the distance formula to find the distance between S and R
:biggrin:

Reply 3

Alex1397

For the first one my friend said that I had to factorise the curve to get (x+1) (x-4) then them equal to zero so then I got x = -1 and then x = 4. Then for the co ordinates for p was (-1,0) and q was (4,0).
Then I differentiated the curve to get dy/dx = 2x-3 I'm not sure what to do next

The second question i manage to differentiate the curve to get dy/dy = 2x+k

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