What is the eq of the line tangent to the parabola 2x^2 - 3x + 2y - 4 = 0 at pt(-2,-5)?

Question

Show a solution pls.

Answer 1

-2y = 2x^2 - 3x - 4
y = -x^2 + (3/2)x + 2
Using Calculus
y' = -2x + 3/2
y' = -2(-2) + 3/2
y' = -5/2 (slope)
y=mx+b
-5 = (-5/2)(-2) + b
-5 = 5 + b
b = -10
y = (-5/2)x - 10

Answer 2

bourqueno's calculation of slope is also incorrect, saying that
-2 * -2 + 3/2 = -5/2. Everything else there is correct.

rewriting equation:
y = (-2x^2 + 3x + 4) / 2
= -x^2 + (3/2)x + 2

-(-2)^2 + (3/2)*-2 + 2 = -4 -3 +2 = -5, so the point is on the parabola

We take the derivative to get the slope of the tangent line through the point:

y'(x) = -2x + 3/2

inserting x = -2:
y'(-2) = -2*-2 + 3/2
= 4 + 3/2
= 11/2

So the slope of the tangent line at this point is 11/2.

To get the equation of the line passing through the point, we can calculate using the slope intercept formula as it was done in previous answers. We can also look at the relation between an arbitrary point (x,y) and the point (-2,-5), and get the equation directly:

(y - (-5)) / (x - (-2)) = 11/2
(y + 5) / (x + 2) = 11/2
y + 5 = (11/2) * (x + 2)
y + 5 = (11/2)x + 11
y = (11/2)x + 6

This is the final equation of the line.

Answer 3

Rewriting x+2y - 4 =0 as y = - (a million/2) x + 2, we see that this line has slope -(a million/2). The perpendicular line will, subsequently, have slope 2, and its equation would be: y = 2x+b. The latter could bypass via (4,5), so 5 = y = 2(4)+b, and so b=-3. to discover the x-coordinate of F, we set -(a million/2)x + 2 = 2x-3. fixing for x supplies x = a million/5. The y could be discovered from the equation of the two line.

Answer 4

Answerer #2 (bourqueno77) is correct. The other folks made some typos and got the wrong answer.

Correct answer is y = (-5/2)x - 10

Answer 5

2x^2 - 3x + 2y - 4 =0
check before starting
2(-2)^2-3(-2)+2(-5)-4=0
8+6-10-4=0
0=0
2y=-x^2+3x+4
y=-.5x^2+1.5x+2
y'=x+1.5
y'(-2)=-2+1.5=-.5
line has slope -.5 & includes the point (-2,-5)
y=-.5x+b
-5=-.5(-2)+b
-5=1+6
b=-6
line is
y=-.5x-6

Answer 6

David S got it right.

Answer 7

2x^2-3x+2y-4=0
4x-3+2dy/dx=0
2dy/dx=-4x+3
dy/dx=-2x+3/2
at x=-2
dy/dx=4-3
=1
equation=y-5=1(x+2)
=x-y+7=0