Coordinates for 2 x tangents for hyperbolic curve???

I'm looking for other coordinates on the curve where the tangent is parallel to y=bx+3. I have solved for a, b, c and tangent equation.

Question.

(-3,a) is on curve y=5x-7/2x+10. The tangent at (-3,a) is parallel to y=bx+3 and cuts the y-axis at (0,c).

How do I get the __other coordinates__ on the curve where the tangent is parallel to y=bx+3?

I’m sure I need to use the gradient perpendicular to the tangent (m=-1/4) and set it equal to y=5x-7/2x+10 or y`equation and solve for x?

Any help would be appreciated.

Workings

I have solved a,b,c (-5.5,4,6.5) and the tangent equation y=4x+6.5. But graphing the curve shows me there is 2 hyperbolas so I need the points on the curve for the other tangent equation.

1. st x = -3 into y=(5x-7)/(2x+10) to get -5.5=a ;. (-3,-5.5).

2. find y` of y=(5x-7)/(2x+10). Use quotient rule..vu`-uv`/v^2.

Then you get [5(2x+10)-2(5x-7)]/[(2x+10)^2]. Then st x =-3..y`=4.

3. y`=4 is the gradient :. b=4 in y=bx+3.

4. Find equation for tangent at (-3,-5.5,m=4) using y-y1=m(x-x1). Becomes y=4x+6.5. As y=mx+c:….c=6.5.

Hence

A=-5.5, b=4, c=6.5 and tangent equation is y=4x+6.5

y=(5x-7)/(2x+10) is correct not ........

correction to c and tangent equation

y=4x**-6.5**. As y=mx+c:….**c=-6.5**. [/size][/font]

Hence

A=-5.5, b=4, **c=-6.5** and tangent equation is __y=4x-6.5__

__should be__

Becomes y=4x**+6.5**. As y=mx+c:….**c=6.5**. [/size][/font]

Hence

A=-5.5, b=4, **c=6.5 and tangent equation is y=4x+6.5**

still unsure about the other coordinates?

**The coordinates of the other point on the curve where the tangent is parallel to y=bx+3.** The answer is (-7,10.5).

I'm still unsure how to get -7. I know that if I s.t. y=10.5 into y=(5x-7)/(2x+10) I get -7.

How do I get the -7?